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Rope DriviDff. 

A Treatise on the Transmission of Power by means 
of Fibrous Kopcs. Nearly loo figures. i7nio, 
cloth, ^.oo. 

nt of Power. 

rs, and the M 
76' figures, izmo, cioth, f^.oo. 

A Treatise on Steam-boilers. 

Their Strength. (3onst ruction, and Economical 
Working, liy Robert Wilson. C.K. Enlarged and 
Illustrated from the fifth English edition by J. J. 
Flather, Ph B.. Purdue Tniversity. 108 fine illus- 
trations, tables, etc. lamo, cloth, 

** Wilson's work on steam-boilers has become 
standard amongst practical men both here and in 
England. The original work has been slightly altered 
by the American author with Mr. Wilson's permisFionj 
IS very complete and one that can be recommended. 
— Affirrican Machinist. 


O " 










Jnvfessor of Meehanieal Engineering, Purdue UniverHty. 





» t J 



liOl^POl^: CHAPMAN ^ HALL, Limjtbo, 




534700 A 




The following treatise has been prepared to supply the 
existing need of a comprehensive manual of practical in- 
formation concerning rope-driving and the principles upon 
which the practice rests. 

In its preparation free use has been made of whatever 
literature could be found relating to the subject, and refer- 
ences for further investigation are given in foot-notes 
throughout the work. 

Most of the data, however, have been collected by the 
writer, who desires to acknowledge his indebtedness to 
those who have assisted in the work by furnishing infor- 
mation and drawings. Especially to be mentioned are Mr. 
C. W. Hunt, the well-known authority upon this subject; 
and Mr. Spencer Miller, long identified with the practice 
of rope transmission. 

J. J. Plather. 

Lafayette, Ind., Oct. 1895. 





Introduction 1 

Leather belts and spur-gears — Early use of ropes — Advan- 
tages of ropes — Losses in various systems of transmission. 


Multiple Rope System 10 

Rope- wells — Distribution of power to several floors — Influ- 
ence of arc of contact — Rope-splicing. 


Continuous Rope or Wound System 25 

Vertical transmission through several floors — Tension-car- 
riages— Rope-tightener — Contraction of ropes — Transmission 
at an angle — Side lead — Multiple idlers— Dynamo driving — 
Jack -shafts — Coil- friction — Winder-pulleys — Application of 
ropes to water-wheels — Rochester plant. 


Long Distance Transmission 68 

Hirn's use of steel band — Wire ropes — Use of shafting- 
Limit of length for shafting — Draw-rods — Efficiency- of 
power transmissions. 

Fibrous Ropes ... 75 

Rawhide— Leather — Manilla — Cotton — Structure of cotton 
fibre — Strength of cotton ropes— Manilla fibre. 


Manufacture op Ropes 85 

Size of yarns — Degree of twist — Lambeth rope — Lubrication 
of ropes — Stevedore rope — Effect of tar on ropes — Area of 
rope — Strength of manilla ropes — Factor of strength and 
^ear — Working strain — Size of ropes, 


Weak op Ropbs 103 

Influence of pulley diameter — Internal wear — External wear 
— Harmonic vibration— Life of ropes — Weiglit of ropes. 

HoHBE-POWEiR Transmitted BY Ropes ', Ill 

Coefficient of friction— Difference In tensions— Influence of 
centrifugal (ores — Graphic representation — Relative cost of 



TUe catenary — Approximate equations— Table of deflectic)n8 
— Method of laying out curve — Inclined transmissions— Ten- 
sion-weight for given deflection. 



Engine- friction— Effect of temperat are— Friction of shafting 
— Power al>sorbed by shafting. 


LoasBB IN llOFE-DRfvisa—Goneluded 159 

Resistance due to l>ending — Wedging in the grooves— Effect 
of groove angle — Wood rims— Differential driving — Creep 
of ropes. 



Least diameter of pulley — Uniformity of pitch — Milled 
grooves — Cast grooves — Light pulleys — Wood filled pulley 
rims— Rim sections — Proportions for groove — Idlers— Hutra 
— Doublearms— Split pulleys — Diameter of bolts— Sections 
of anns^Stresses in pulley — Methods of Joining arms and 
rim — Built-up rope-pulleys. 




1. Limit of Length for Steel Shafting 70 

IL Limit of Length for Wrought-iron Shafting 71 

III. Strength of Cotton Transmission Ropes 81 

IV. Strength of Manilla Transmission Ropes 95 

V. Executed Rope Transmissions 96 

VI. Greatest Revolutions per Minute for given Diameter of 

Rope 104 

Vn. Weight of Ropes * 110 

VIII. Values of 1-e for a Working Stress Equivalent to 200d* 

pounds 117 

IX. Angle embraced by Rope 117 

X. Friction and Stress Moduli 118 

XI. Horse-power Transmitted by Ropes 121 

XII. Relative First Cost of Rope-driving 128 

XIII. Relative Wear and Cost of Rope per Horse-power Trans- 

mitted 125 

XIV. Deflection of Rope 134 

XV. Friction Per Cent under Varying Loads 143 

XVI. Power-absorbed by Friction in Line-shaft 152 

XVII. Power absorbed by Friction in Jack or Head Shafts .... 154 
XVIII. Power absorbed by Lead-shaft carrying High-speed 

Ropes 156 

XIX. Angle of Grooves for Equal Adhesion 169 

XX. Values of <?»' and f F 180 

XXI. Least Diameter of Pulley for given Diameter and Speed 

of Manilla Rope 180 

XXn. Rope-pulleys for General Work 181 




Although toothed gearing and belts are the most fa- 
miliar mediums for the transmission of power in mills and 
factories, systems of rope transmission for this purpose 
have been in use for many years, but it is only within the 
last ten years that they have given promise of being gen- 
erally recognized in this country as a convenient and effi- 
cient means of accomplishing the ends for which they were 
designed. Until about fifty years ago it was generally 
thought by engineers that cotton- and woollen-mills, and 
all others requiring a considerable amount of power, could 
not be run effectively without large and ponderous lines of 
upright and horizontal shafts of either cast or wrought iron 
and heavy trains of gear wheels. 

When large leather belts began to be introduced as a 
substitute for gears it was thought to be an experiment of 
very doubtful result, if not altogether impracticable;* but 
when high speeds and lighter shafting were used in con- 
junction with the wide belts, the marked success which at- 
tended their general adoption in America during the next 
twenty years attracted considerable attention in England. 
It was not, however, until about thirty years ago that the 

* Journal Franklin Imt, 1837. 



American system began to replace the old-fashioned gear- 
ing. The belt system made very slow progress in England, 
however, and before it had been at all extensively adopted 
a newer method was introduced and quickly came into 
prominence, making such rapid progress as to almost en- 
tirely supplant the old wheels. 

The use of hemp rope for power transmission had been 
revived about 1860 by the Messrs. Combe & Barbour of 
Belfast, who introduced it successfully into small mills in 
the north of Ireland. This was followed by its speedy 
adoption in the jute-mills of Dundee, and subsequently in 
the cotton-mills of England. 

Previous to this, fibrous ropes had been in use for trans- 
mission purposes, but their application had been limited. 

At the Colonial Rope Factory at Great Grimbsy, in Lin- 
colnshire, from 1830 to 1837, ropes had been employed for 
taking off the power from the engine and communicating 
to the first motion shaft. The plan was very commonly 
adopted in connection with rope-works, where the driving 
rope employed was known as the fly-rope for working the 
equalizer from end to end of the ropewalk. The machin- 
ery in connection with the flax-spinning mills at the same 
place was also driven by means of rope gearing.* 

Ropes were also used in this country many years before 
their more general adoption in England which followed the 
movement introduced by Messrs. Combe & Barbour. In a 
recent communication, Mr. Samuel Webber states that he 
remembers the occasional use of rope-driving for temporary 
purposes *'back in the forties.'' In one case he mentions, 
power was carried from a small engine outside through a 
window into a mill to grind cards before the wheels and 
main belts were ready. 

The nse of ropes, however, was not common, and it has 

^M^^_ I — — , __ . - ■ — —^ 

Proc I. M. E. 1876. p. 893. 


only been in recent years that rope-driving has come into 
prominence as a factor in power transmission. 

By this system, according to one of its earlier promoters,* 
Mr. Jas. Durie, large powers were transmitted "by means 
of round ropes working on grooved wheels, which in some 
parts of this country [England] have been largely sub- 
stituted for toothed gearing. In this mode of driving, the 
fly-wheel of the engine is made considerably broader than 
the fly-wheel of an engine having cogs on its circumference; 
and, instead of cogs, a number of parallel grooves for the 
ropes are turned out, the number and size of which are 
regulated by the power to be taken off the fly-wheel. The 
power which each of the ropes will transmit depends upon 
their size and the velocity of the periphery of the fly-wheel.^' 

As rope-driving has, until recent years, been a matter 
largely of experiment, the results which have been obtained 
from its use have not always been of uniform excellence, 
mainly, however, because designers have failed to properly 
recognize the requirements. 

As the conditions under which the systems were installed 
have been so varied, it is not surprising to find many cases 
where the ropes have been rapidly worn out and replaced 
by leather belting, or other methods of transmission ; bu t 
where rope-driving has been tried and has failed, investi- 
gation will invariably show the absence of suitable condi- 
tions or a disregard of correct principles of design or con- 
struction. In many applications too great a strain is put 
upon the rope, and the stretch and wear are rapid; in ad- 
dition to this, the pulleys are often of unsuitable size, and 
the rope is unnecessarily weakened through the action of 
the fibres upon one another. Both of these causes are a 
constant source of annoyance in a rope-drive which has 
been poorly designed or constructed. A case in point can 

*?roc. I. M. B. 1970. 


be shown where a rope would stretch to such an extent 
that it had to be taken up every few days when new, and 
later every two or three weeks — a rope under these condi- 
tions lasting less than four months. Investigation showed 
that a tension-weight of 600 pounds had been placed 
upon the rope, which was of i inch diameter. A weight 
of 50 pounds was substituted, and the rope has since been 
running satisfactorily for three years, and has only been 
taken up once in that time. 

Cases where ropes have suddenly broken are few in num- 
ber, the risk in this respect being reduced to a minimum 
by the fact that any defect in a rope, arising either from 
wear or other cause, will show itself long before the point 
of danger is reached. 

The ropes by which the power is transmitted consist of 
an elastic substance, and their lightness, elasticity, and 
comparative slackness between the pulleys are highly con- 
ducive to their taking up any irregularity that may occur 
in the motive power. 

Their quiet working and convenience in application, 
much more so than wide belts, are also features which 
caused ropes to be looked upon with great favor. An- 
other reason for its rapid progress in England, which was 
considered a great advantage by the millowners, was the 
adoption of the multiple rope, now known as the English 
system, in which a number of single ropes are spliced and 
run side bv side. 

The entire freedom from any risk of a breakdown or 
stoppage of the works which might occur with gearing was 
an important factor in replacing the latter by the newer 
system; the working stress in the ropes being but a fraction 
of their breaking strength, any signs of weakness in an in- 
dividual rope would allow it to be removed, and the engine 
run with the remaining ropes until a convenient opportu- 
nity ofEered for the replacement of the weak member. 


One of the great advantages of rope-driving over gearing 
lies in the steady motion produced, but this has been at- 
tributed more to an accidental combination of heavy fly- 
wheel and high velocity than to any inherent advantage in 
the system itself. 

In spur gears diameters of 20 to 24 feet were usual in 
large engines, while for ropes the pulleys are 25 to 35 feet 
in diameter; and, whereas the gears weighed from 20 to 25 
tons, we find rope pulleys used for the same work weighing 
from 60 to 75 tons.* 

When we consider the speed at which these heavy wheels 
are run — from 3000 to 6000 feet per minute — it is not 
surprising that uniform rotation is obtained; and whether 
it be that the energy stored in the moving mass prevents 
fluctuation, or whether the elasticity and other properties 
of the rope perform the same office, or — which is more 
reasonably the case — all of these factors act together, the 
truth remains that steadier running and a greater output 
are now obtained with rope-driving than was formerly the 
case with toothed gearing. 

The general experience is not altogether in favor of ropes, 
for, while the advantages of smooth running and easy 
handling are conceded, it is also acknowledged that the 
extra weight and greater width of pulleys increase the 
journal friction over that found with toothed gearing, and 
that otherwise a greater loss of power occurs, the causes of 
which will be discussed subsequently. 

* The Walker Manufacturing Co. of Cleveland recently made four 
large rope pulleys for the Broadway Cable Railway Company, New 
York, which weighed 104 tons each. These were 32 feet in diameter 
and were grooved for thirty-four 2-inch ropes. 

Another example of heavy rope wheel is given in London Engin- 
eer, January 11, 1884. This wheel was made by Hick, Hargraves & 
Co, Bolton, England, for a cotton mill in India, and is 80 feet in 
diameter, 15 feet face, grooved for 60 ropes to transmit 4000 horse- 
power. Its weight is 140 tons. 


It is difficult, however, to determine the relation between 
the power absorbed by ropes and gears, for in nearly all 
eases where rope-driving has been substituted for gears, 
other changes have been made at the same time, or the 
engines were, after the alteration, driven at an increased 
speed, so that there has been little opportunity for com- 
parison. There is, however, a. generally- accepted opinion 
among engineers that the loss in rope-transmission is from 
5 to 10 per cent greater than with gearing. 

Mr. A. G. Brown* states that in the older cotton-mills of 
England, where the main drives are by gearing, and belting 
is used for the intermediate and machine driving, the fric- 
tion of the engine, shafting, gearing, and belting averages 
about 20 per cent of the whole power, the engine develop- 
ing at full loads from 500 to 800 I. H. P. 

These engines were compound condensing, and consisted 
in each case of an overhead beam engine Avhich had been 
converted into a compound by the addition of a high-press- 
ure cylinder between the crank and beam centre. 

In the newer mills, for doing practically the same work, 
and where the main and some of the intermediate drives 
are by ropes, the friction of the engine, ropes, shafting, and 
belting averages 23 to 25 per cent, of the whole power, the 
engines developing from 800 to 1500 1. H. P. It is fair to 
assume that the newer engines have at least no greater per- 
centage of friction than the older ones, except that due to 
an increased journal friction attributable to the larger 
journals and greater weight of the rope-pulleys ; also the 
friction of the intermediate shafting and small belts should 
be the same in each case ; therefore it is reasonable to con- 
clude that the increase of power absorbed by the use of 
rope-driving is chargeable to the system itself. The writer 
has assumed in similar cases that the loss at the engine in 

* American Machinist ^ July 21, 1888. 


rope driving is about 10 per cent of the indicated horse- 
power, that an additional 10 per cent is absorbed by the 
mill-shafting, and that from 5 to 8 per cent may be at- 
tributed to losses in the rope itself due to resistance to 
bending, wedging in the grooves, differential driving effect, 
and creep, all of which' affect the loss to a greater or less 
extent. As compared with the above, the friction of shaft- 
ing and engines in American cotton-mills, where belting is 
used exclusively, indicates that the percentage of loss 
where large belts are employed is probably a trifle less than 
that obtained with ropes in English mills ; but the condi- 
tions of practice are so varied that it is difficult to compare 
the two systems from published results of tests. As shown 
on page 155 for similar installations, the loss absorbed in 
shaft-friction will not materially differ in the two systems ; 
for large transmissions the engine -friction should be less 
with ropes, but the losses in the ropes themselves due to 
slip, differential driving, bending, creep, and other causes, 
may, without special precautions, exceed to a small degree 
the losses due to the belt. 

Mr. J. T. Henthorn, in a paper read before the American 
Society of Mechanical Engineers, states that the friction of 
the shafting and engine in a print- mill should not exceed 19 
per cent of the full power; but out of fifty-five examples of 
a miscellaneous character which he has tabulated only seven 
cases are below 20 per cent, 20 vary from 20 to 25 per 
cent, fifteen from 25 to 30 per cent, eleven from 30 to 35 per 
cent, and two above 35 per cent. We note that the greater 
number lies between 20 and 25 per cent; allowing a varia- 
tion of 5 per cent each side of these limits we shall obtain 
values frotn which a fair average may be determined. This 
will include those cases under 20 per cent, but not those 
over 30, from which we find the mean loss to be 23.9 per 
cent of the total power. 

Mr. Barrus, speaking of this subject, quotes eight cases. 


the data of which were obtained from tests made by him- 
self in various New England cotton-mills, in which the 
minimum percentage was 18 and the maximum 25.7, the 
total average being 22 per cent. 

Mr. Samuel Webber states that 16 per cent of the total 
power of a mill is sufficient to overcome the friction of 
shafting and engine — 10 per cent for the shafting and 6 
per cent for the engine. But in this estimate Mr. Webber 
does not include the loss due to the belts running upon 
loose pulleys, which he does not consider to be part of the 
shafting, as they are not so running while the machinery 
is in operation ; and when it is not, they may be thrown off 
as well as not, except for convenience. 

He further estimates, both from his own experience and 
the observations of others, that the power consumed by the 
machine belts on the loose pulleys in a large cotton-mill 
is about 8 per cent of the whole. This 8 per oent 
added to the 16 per cent loss due to shafting and engine 
will give 24 per cent of the total power — a result which 
agrees closely with the average values given above.* 

Considering the greater loss which occurs in the use of 
ropes and belts for main drives, the recent revival of gear- 
ing for this purpose in England has much in its favor. 
Of this Mr. Geo. Kichards states :f "The advantages of 
rope transmission for main drives in large plants would 
not be as apparent if compared with modern gearing. 
The kind of gear used for this purpose to-day is not 
the rough cast gear used formerly, whose uneven motion 
produced a rumbling which could be heard a mile or two 
from the mill. The present gears are often machine-cut, 
made to bear equally on each tooth, and with a contact 

* See ** Dynamometers and the Measurement of Power," John 
Wiley & Sons, New York. 

f Richards's "Mechanical Progress,** 1891. 


across the whole face of the gear, causing little more noise 
than the ropes. 

With greater speed and stronger and heavier gears steady 
running is insured, and by using properly-proportioned 
machine-cut teeth comparatively little noise results. The 
saving of space is also an important factor advanced in 
favor of gearing." At the present time steel gears 30 feet in 
diameter are being made, with all the teeth cut, for trans- 
mitting the powers of mill-engines in the Oldham district, 
while in this country machine-cut gears from 30 to 50 feet 
in diameter are in use. 

However, although such gearing may be very superior 
to the former slow-running cast gears, it is questionable 
whether it can ever produce the same steadiness of running 
which is so largely a distinguishing feature of rope-trans- 
mission. Any shock or sudden fluctuation of load must 
necessarily be transmitted through the gear teeth, whereas 
with rope transmission such shock is partially absorbed by 
the more or less elastic ropes and subsequently given out 
by their recoil. 

For this reason when uniformity of speed is desired 
ropes are generally to be preferred to gears, even when the 
latter are working under their most advantageous con- 



Where ropes have been used to replace gearing in the 
English mills the plan adopted has been to put in a new 
grooved fly-wheel, or to place grooved segments upon the 
existing fly-wheel, when the speed could be increased suffi* 
ciently to allow of a limited number of ropes being em- 
ployed, and the width of the wheel-pit was also sufficient 
for the purpose ; but if this plan could not be adopted 
grooved pulleys were put on the intermediate shaft, and the 
ropes carried to the different stories of the mill. It has 
sometimes been necessary to put in a countershaft, so as to 
gain speed and obtain a sufficient distance between the 
centres of the shafts on which the pulleys are placed. 

Where rope-driving has been installed in new factories 
special provision has been made for the ropes, and we find 
in such cases rope-wells or chambers built in, suitably fitted 
with platforms and staircases to give access to pulleys and 
bearings on the various shafts, as shown in Figs. 1 and 2. 

The majority of drives are arranged so that the ropes are 
horizontal or inclined rather than vertical, and with the 
driving or tight side of the rope on the lower side of the 
pulleys ; then, when transmitting power, the two sides ap- 
proach each other, and the arc of contact is increased. 

An additional advantage is that obtained by the weight 
of the rope acting on both pulleys, thus allowing a low 
initial tension to be maintained. This does not hold for 
short distances between centres, as under such conditions 
the weight of the rope adds little to the total tension; on 
the other hand, where the distance between the pulleys in 



vertical drives is small, the relative weight of the rope be- 
ing small as compared with its tension, there will be little 
tendency for the rope to leave the bottom sheave. 

With these conditions the eflSciency of vertical drives 
will a])proach that of a horizontal or inclined arrangement 
of ropes. 

The manner of distribution of the power to the several 
floors of a mill is shown in Fig. 1, which represents a 
plant designed by Messrs. Lockwood & Greene of Boston 
for the Lanett Cotton Mills, West Point, Ga., in which 
1100 h. p. is delivered from the engine by means of twenty- 
six IJ-inch ropes. The fly-wheel is 2G feet in diameter, and 
makes GO revolutions per minute, corresponding to which 
the velocity of the rim is 4900 feet per minute. As will be 
seen from the figure, the driving-sheaves are placed in a 
well in the middle of the factorv, and the line-shafts ex- 
tend to the right and left, as shown. 

There is no line-shaft on the second floor, as the various 
machines may be driven from below. 

The distribution is as follows : 

Rope-drives in Lanett Cotton Mills. 

1st floor. 
3d floor. 
4tli floor 


Dia. of 

Dia. of 
















of Pullej' 

per minute. 




A similar plant, also designed by Messrs. Lockwood & 
Greene, has been recently erected at the Naumkeag Cotton 
Mills, Salem, Mass., in which 1800 horse-power is distrib- 
uted to five floors by means of forty-one IJ-inch mnnilla 
ropes.* The fly-wheel is 26 feet in diameter, by about 9^ 

* Power, March 1895. 




feet face, and weighs 150,000 pounds. The velocity of the 
ropes is the same as in the previous instance, namely, 4900 
feet per minute. The distribution is as follows: 

Number of 

Dia of 

Speed of Rope. 


l»t floor 






If inches. 


4900 ft p. m. 
« i 



2(1 '• 

M '* 

4th " 

5th " 





A system of main driving-gear designed and erected by 
Jolin Musgrave & Sons at the Atlas Mills, Bolton, Eng., is 
shown in Figs. 2 and 3. Tliis mill is 300 feet long by 135 
feet wide, with a shed on one side 335 feet long by 45 feet 
wide, and contains 84,000 spindles. 

The engines are tandem compound, 24 and 46 by 6 ft. 
stroke, and run at 50 revolutions per minute. The average 
horse-power is 1050. The rope-wheel is 32 feet in diameter 
and is grooved for thirty-two l|-iuch cotton ropes, which 
run at 5026 feet per minute. 

The arrangement of shaft is as follows: On the ground- 
floor there are lines of shafting, the main shaft being 
driven from the rope-drum by means of ten ropes If inches 
diameter running on a pulley 9 feet 4f inches in diameter, 
which runs 170 revolutions per minute. 

On the main shaft, close to the wall of the rope-well, is 
a pulley 6 feet diameter, grooved for four If -inch ropes, 
which drives, through a similar-sized pulley, the line-shaft 
on the right. These ropes run at a velocity of 3205 feet 
per minute; also on the main shaft, but on the opposite 
side of the rope-well, is another pulley, 8 feet diameter, 
grooved for six IJ-inch ropes, driving on to a pulley 64^ 



inches diameter on the first line-shaft to the left of the 
main shaft. This shaft runs 250 revolutions per minute, 
corresponding to which the speed of the ropes driving it is 
4274 feet per minute. 

The second line-shaft on the left is driven from the first 
one by means of a pair of pulleys 4 feet diameter, grooved 
for five If -inch ropes. This second shaft runs 250 revolu- 
tions per minute, and the ropes driving it have a velocity 
of 3140 feet per minute. 

The second line-shaft, mentioned above, drives the line- 
shaft in the shed by means of a pair of pulleys 4 feet di- 
ameter, grooved for four l^-inch ropes, driving the counter- 
shaft shown on plan, on which is a pulley 3 feet 4 inches 
diameter, grooved for five l:J^-inch ropes, driving on a 3-ft. 
pulley on the shed line-shaft. These ropes have a velocity 
of 2640 feet per minute and give to the shaft 278 revolu- 
tions per minute. 

All of the shafts described are on the ground -floor of the 
mill. Of the shafts above this, on the next two floors, the 
line-shafts eacli have pulleys 6 feet in diameter, grooved for 
seven If -inch ropes, driven from the main rope-drum. 
These shafts run at 266 revolutions per minute. The 
shaft on the upper floor also runs 266 revolutions per min- 
ute and is driven from the main rope-drum through a pul- 
ley 6 feet in diameter grooved for eight If-inch ropes. 

The distance from the centre of upper shaft to the cen- 
tre of crank-shaft is 89 feet, and the length of each rope re- 
quired for this drive is about 250 feet. 

Another arrangement of rope-drive for cotton-mills is 
shown in Fig. 4, which represents a section through the 
engine-room at the thread-mills of the Nevsky Cotton- 
Spinning Co., St. Petersburg.* In this drive there is no 
rope-chamber, as the whole of the rope gear is contained 

* John Musgrave & Sous, engineers. 


in the engine-room situated in the centre of the mill, which 
is 680 feet long and 90 feet wide. 

There is a short staircase from the engine-room floor to 
the first landing, and the landings above this are reached 
by an ornamental spiral stairway as shown. 

There are two shafts, one from each side of the engine- 
room; these are driven by a pair of right- and left-hand 
tandem compound engines, 30 and 52 by 6 ft. stroke, run- 
ning at 50 revolutions per minute. 

The average power developed by each engine is 1100 

The rope-drums are each 30 feet in diameter and weigh 
62 tons; these are grooved for twenty-eight If cotton ropes, 
which run at 4700 feet per minute. 

The first-and second-floor shafts make 300 revolutions 
per minute, and are each driven by nine If -inch ropes from 
the rope-drum running over pulleys 5 feet in diameter on 
the shafts. 

The shafts on the third and fourth floors run 200 revo- 
lutions per minute, and are driven by five ropes, each If 
inches diameter, running over 7 feet 6 inch pulleys on the 

In each of the two upper rooms there is a second line- 
shaft, driven from the main line-shaft on each floor by 
means of 54-iiich pulleys, grooved for four 1^-inch ropes, 
which have a velocity of 2826 feet per minute. The 
distance from the centre of the upper shaft to the 
centre of the crank-shaft of engine is 56 feet 6 inches. 
This is a short drive for a mill of this size; in fact, all of 
the drives are short, the lower one especially so, being only 
30 feet between centres, the peculiar arrangement of the 
engine-room not admitting of a greater length; but the 
plant is said to work extremely well. 

In these examples the multiple-rope system is used, each 
wind consisting of a separate rope stretched around the fly- 



wheel and its individual shaft-pulley, then spliced. The 
degree of tightness will depend upon the material of the 
rope and the amount of tension in the slack part necessary 
for adhesion. In the majority of cases the initial tension 
is very small compared with the strength of the rope — es- 
pecially so where the horizontal distance between driving 
and driven pulleys is great, as, under such conditions, the 
tension in the slack side due to the weight of rope in the 
huM«.»iiu^ catenarv is often sufficient to prevent slipping; a 

sla^k upper rope in horizontal 
or inclined drives will also 
increiise the arc of contact, 
thereby increasing the grip 
of the rope. 

This is shown in Fig. 5, 
which represents two pulleys 
of equal diameter, arranged, 
as in the upper figure, to 
drive with the slack side uppermost, and, in the lower 
figure, with the slack side below. 

The ditferenco in the arc of contact, as shown in the 
figures, is GO degrees, which would under similar con- 
ditions, with a velocity of 4000 feet per minute, produce a 
difTerence of over ^^r) per cent in the amount of horse-power 
transmitted by the two ropes under the usual working 

New cotton ropes are often stretched as taut as possible 
on account of their extensibility, as they will soon become 
slack enough for good working, and may even have to be 
respliced before becoming permanently set. 

It is the practice of some engineers to strain both manilla 
and cotton ropes as much as possible and unite the ends 
with a temporary short splice when first put oyer the pul- 
leys; after running a few days a permanent stretch is given 
to the rope, which is then respliced with a long splice, the 

Fm. 5. 


strain on the rope being very much reduced in this latter 

The splice in a transmission rope is not only the weakest 
part of the rope, but is the first to fail when the rope is 
worn out. If the joint is not strong the rope will fail by 
breakage or pulling out of the splice, the projecting parts 
will wear on the pulleys, and the rope fail from the cutting 
off of the threads. Formerly much trouble was experienced 
in this way on account of improper splicing. One form of 
joint, according to Cromwell,* was made by pressing the 
ropes firmly together and winding about with stout small 
rope. The spliced part is taken as long as possible in order 
to bend properly over the pulleys and give the required 
strength. As this form of joint made the rope larger in 
diameter at the splice, the effect produced was to run 
faster when passing over the driving-sheave and slower 
over the follower; the resulting motion was very irregular, 
and the wear at the splice rapidly destroyed the rope. 

A very simple splice is sometimes used with rope-driving 
formed by opening out the ends of the rope for 12 or 15 
inches and tying together the individual rope-yarns one 
by one, allowing the ends to lie straight, and serving the 
whole with spun yarn. 

Similar joints wrapped with raw-hide belt-lacing give a 
very smooth splice which lasts well. 

Some engineers favor a short splice, in that it is easily 
made and holds well, and offers a lesser length of enlarged 
portion for surface contact with the pulley. 

If properly made, however, there need be no enlarged 
portion, and since a long splice is stronger we find such 
joints preferred in most cases. 

There are several kinds of long splices varying in length 
from 60 to 80 diameters of rope, but the one which seems 

* J. H. Cromwell, '* Belts and Pulleys," John Wiley & Sous. 



to give the best results in practice is the " English splice, 
directions for which are given* in various trade publications. 
The successive operations for splicing a l|-inch rope by 
this method are as follows : * 

1. Tie a piece of twine, 9 and 10, Fig. 6, around the rope 
to be spliced about six feet from each end. Then unlay 
the strands of each end back to the twine. 

2. Butt the ropes together and twist each corresponding 
pair of strands loosely, to keep them from being tangled, 
as shown at {a), Fig. 6. 

3. The twine 10 is now cut, and the strand 8 unlaid and 
strand 7 carefully laid in its place for a distance of four 
and a half feet from the junction. 

4. The strand 6 is next unlaid about one and a half feefc 
and strand 5 laid in its place. 

5. The ends of the cores are now cut off so they just 

6. Unlay strand 1 four and a half feet, laying strand 2 
in its place. 

7. Unlay strand 3 one and a half feet, laying in strand 4. 

8. Cut all the strands off to a length of about twenty 
inches, for convenience in manipulation. The rope now 
assumes the form shown in {b), with the meeting-points of 
the strands three feet apart. 

Each pair of strands is now successively subjected to the 
following operations : 

9. From the point of meeting of the strands 8 and 7 
unlay each one three turns; split both the strand 8 and 
the strand 7 in halves, as far back as they are now unlaid, 
and the end of each half strand " whipped " with a small 
piece of twine. 

10. The half of the strand 7 is now laid in three turns, 
and the half of 8 also laid in three turns. The half strands 

* From •' Manilla Rope/' C. W. Hunt Co., New York. 


Pig. 6,— Splice fihi H-inch 4-stbakd Bopb. 


now meet and are tied in a simple knot 11, (c), making the 
rope at this point its original size. 

11. The rope is now opened with a marlinspike, and the 
half strand of 7 worked around the half strand of 8 by pass- 
ing the end of the half strand through the roj>e, as shown, 
drawn taut, and again worked around this half strand 
until it reacrhes the half strand 13 that was not laid in. 
'I'his half strand V\ is now split, and the half strand 7 
drawn tii rough the opening thus made, and then tucked 
under the two adjacent strands, as shown in (</). 

\:t. 'Die otiier half of the strand 8 is now wound around 
the other half strand 7 in the same way. After each pair 
of strands has been treated in this manner, the ends are 
vAii otT at \)l, leaving tiiem about four inches long. After 
a few days' wear they will draw into the body of the rope 
or w<;ar olT, so that the locality of the splice can scarcely 
he detcjcted. 

For a three-strand roi)e of the same size the foregoing 
method is slightly modified. After tying the twine 9 and 
10 around the ro2)e about G feet from each end, unlay the 
strands hack to the twine, bring the butts together, and, 
as in Fig. 7, twist the corresponding strands loosely together. 
Now cut twine 10, and unlay strand 8 for a distance of 
four and a half feet from the junction, and lay in strand 
7. Unlay strand 1 four and a half feet, lay in strand 2, 
and cut all the strands oif to a length of about 20 inches, 
as before explained for convenience in handling. The 
splice now assumes an ajipearance similar to (b) with the 
exception that there are only three meeting-points of the 
strands, and these are 4^ feet apart. 

Eafili pair of strands is now subjected to the series 
of opc^rations described for tlie 4-strand splice in steps 9 
to 12 inclusive. 

In splicing a Lambeth cotton rope the operation is modi- 
fi(ul to a still greater extent. 

ROPE-nmviNG. 22a 

Although considered as a troublesome rope to splice, the 
following instructions,* if carefully followed, will enable 
one to make an excellent joint without difficulty. 

1. Tie a piece of twine 9 and 10 around the rope to be 
spliced about 6 feet from each end. Then unlay two 
strands together of each end back to the twine. Butt the 

ropes together and tie each set of strands temporarily, a- 
shown at Fig. A. 

2. The twine 10 is now cut, and the strands 6 and 8 
unlaid together, and the strands 5 and 7 carefully laid in 
their places together for a distance of 18 inches. Then 
nnlay strands 6 and 8, also 5 and 7, and tie strands 6 and 
5 together temporarily. Next unlay strand 7 and lay in 
strand 8 in its place for a distance of 3 feet from strands 6 

"Prepared tor this work by the Manufacturers' Engineering Co., 


and 5. Theu tie strauOs 3 and ', [emporarilT. Next cut 
off Ihe ends of the core so tliat they will butt together. 
Stmnds 1, 2, 3, and 4 are nest laid in the ^ 

strands 5, 0, 7, and S, but in the opposite direction. Care 
must be tiiken lo keep the tarns in the strands, or otherwise 
they will be soft and bulky. Next cut off all the strands 

to a length of about 34 Indies, for convenience in handling. 
At this point the splice shoiikl be as shown in Fig. B. 



The tension strand of a Lambeth cotton rope is the soft 
white yarn running through the centre of the strand, 
and is called the tension strand through its having to 
bear the strain put upon the rope in the transmission of 

The friction bands of a Lambeth rope are the twisted 
outside yarns which are tubed around the tension strands 
to protect them from wear and contact in the grooves of 
the pulley. 

3. Take strand 2 and unlay it two turns and remove the 
ten friction bands, then lay in tension strand 2 back again 

Fig. D. 

one turn, split out i of tension strand 2 and lay in the re- 
maining J of tension strand 2 for one turn. This will 
bring it to its former position. Eemove the ten friction 
bands from strand 1, and tie tension strand 1 and f of 
tension strand 2 in a simple knot. At this point of the 
knot the rope will be its original diameter, as shown in 
Fig. C. 

4. Divide the friction bands removed from strand 1 in 
two parts, and take J of tension strand 2, put it between 
the two parts and over tension strand 1 and through the 



centre of the rope with tho niurlinspike. Next take ten- 
sion strimd 1 and work it around tlie | of tension-strand 2 
in the manner as shown at Fig. D. 

5. Draw it taut and continue to work it around j of ten- 
sion strand 3 until it reaches the ^ of tension strand 2; 
at this point I of tension strand 1 must be removed, and 
continue to work } of tension strand 1 around tension 
strand 3 until it reaches the friction bands removed from 
strand 2; divide these friction bands in two parts, and take 
f of tension strand 1, put it between the two parts and 

Fig. E. 

over tension strand 2, and through the centre of the rope 
with the marlinspike. Next take the quarter of tension 
strands 1 and 2 and pass them through the centre of tho 
rope on opposite sides with the marlinspike. Then half 
of the friction bands should be passed through the centre 
of the rope at each end with the spike. At this point 
the splice is complete, with the exception of cutting off 
the ends, and should be as shown at Fig, E. 

6. The strands 3, 4, 5, 6, 7, and 8 should he worked in 
the same manner as 1 and 2. 

Instead of nsing the ordinary marlinspike it will be 
found very convenient to drill out the body as shown in 



Pig. 8, leaving only a thin shell four or five diameters 
By inserting the end of a strand in the bore of the mar- 


linspike the latter, with the strand, may be passed through 
and around the other strands as desired with much less 
trouble than ordinarily attends the operation. 

Fig. 8. — Improved Form op Marlinspikbl 

For small braided ropes which cannot be spliced, a 
very convenient method of joining the ends is by the 
use of copper ferrules as shown . in Fig. 9, which repre- 

FiG. 9.— Coupling por Braided Rope. 

sents a form of joint devised by Mr. B. Frank Barnes^ 
Eockford, 111. 


Samples of tins splice were furnished by Mr. Jos. 
Buruett, which showed an efficiency of about 85 per cent. 
Thus in two samples the average breaking strength of the 
rope was 380 and 375 pounds respectively, while the splice 
pulled out under a strain of 320 pounds. 

Ill this case the rope was a f-inch braided cotton cord 
which had been in use about three years. The coupling 
consists of a piece of copper tubing 1 inch long, into one 
end of which the rope is inserted about half way. A 
groove is then compressed around the tube and rope, by 
means of a special tool; the open end of the tube is then 
filled with sealing-wax and heated until the wax boils, 
then the other end of the rope is inserted, and the tube 
compressed. The melted wax fills the end of the rope, 
making a solid joint between the shoulders. With the 
large pulleys adopted (32 to 48 inches in diameter) no 
trouble is experienced, and the ropes last from two to 
three years, but the copper ferrules are changed about 
every four months. 

Wood pulleys are used„ and the grooves are filled with 
leather. Some of these ropes run as high as 5300 feet per 
minute. All the ropes are operated on the American or 
continuous wind system. 

E0PE-DRIVI2!fG» 25 


A GOOD example of this system of rope transmission i? 
shown in Fig. 10, which represents a plant designed bv 
Mr. T. Spencer Miller, and erected for the Western Electric 
Company, New York, by the Link Belt Machinery Co. 
In this case vertical ropes are used, which are arranged to 
transmit the power of two 200-h. p. Eussell engines, 
cylinders 18 by 27 inches, making 125 revolutions per 
minute; fly-wheels 10 feet diameter, each turned with 
eight grooves for 1^-inch rope. The ropes are of raw- 
hide and wound continuously around the pulleys. As the 
rope leaves the fly-wheel at the left-hand side it runs over 
an idler, and from thence to a tension -pulley, or tightener, 
which is suspended in such a manner as to be drawn back 
by the weights, as shown. The arms which support the 
tighteners are hung from rollers, which are grooved to fit 
the surface of a section of extra heavy wrought-iron pipe, 
upon which they roll. From this tightener the rope 
passes direct to the right-hand groove of the pulley on the 
main shaft above the engine, the tightener-pulley being 
inclined sufficiently to make the bottom come in line with 
the left-hand, while the top comes in contact with the 
right-hand groove. 

The main pulleys which drive the shaft above the 
engines are mounted upon and keyed to sleeves 10 inches 
diameter, which extend out on each side far enough to 
form journals, by which they are supported in pedestals 
independently of the shaft (see Fig. 33). Through the 
sleeve is a hole considerably larger than the shaft which 
passes through them and which is supported by separate 



pedestals. One end of each sleeve is so formed as to 
make, in connection with a sliding collar which is on the 
shaft, a positive interlocking clutch, which can be thrown 
in or out by a lever. In this way both engines may be 
working at the same time^ or the shaft may be run by 
either engine alone, the other pulley standing and impos- 
ing no friction upon the shaft. All the bearings of this 
shaft are adjustable laterally by set-screw and vertically by 
wedges. The other large pulleys upon this shaft are 
driven by friction-clutches, and are used for driving dyna- 
mos, each pulley driving two dynamos arranged tandem, 
one belt running over the other. This shaft runs at 220 
revolutions and is 4^ inches diameter. At one end of the 
main shaft is a pulley having twelve grooves, in which run 
two ropes side by side to the top of the building and 
around the various pulleys down again. Either of these 
ropes is calculated to be amply strong for the work, but 
two are used to avoid the necessity for stopping should 
one break. Each of them winds three times around the 
pulleys, thus giving six driving-strands. To avoid crowd- 
ing, the tightener for one of these is placed upon the floor 
above the other. The ropes pass from the main pulley 
three times around the pulley above and then go to the upper 
floors, as indicated. From the shafts, the wood-working 
machinery, blowers for the foundry, and some of the eleva- 
tors are operated; the other lines being used for driving 
light machine tools, such as are used in making electrical 

Each floor has a cut-off coupling, which is so arranged 
that in case of accident it can be cut off at a minute's 
notice, or when running overtime any floor can be cut off. 
thus saving the cost of running any more machinery than 
is necessary. 

The use of the tension -carriage plays an important part 
in the American system of rope transmission. As usually 


made it is automatic in its operation and bo weighted as 
to give a constant tension to tiie rope, as indicated in 
Fig. II, In this arrangoment an initial tension is given 

Fio. 11. — Automatic Tightener fob Ropg Trakb: 

and maintained by the automatic ten si on -carriage, which 
is free to move backward or forward on a horizontal track 

Fio. 13.— Tekbion-cabkiagb. 

as the load changes or the rope stretches, always taking 
up the slack and maintaining the proper tension. An- 
other form of tension -carriage is that shown in Fig. 12. 


In tliia case the carriage is mounted on gas-pipe or solid 
shafting, and is provided with ball bearings arranged with 
cast pockets so that the balls are allowed to circnlate 
lengthwise of the bearings. 

An arrangement of light angle-iron tracks supporting a 
four-wheeled carriage is used to a considerahle oxtt nt and 
makes an excellent tightener where it can be 
employed, as it is cheap and readily set up. 

It is obvioDS that vertical tensions may be 
arranged in a similar manner to those shown 
in Figs. 11 and 12. In such cases the weight 
may be suspended directly from the carriage or 
even the sheave, as in Fig. 13, or it may be led 
off in any desired direction either above or 
below the tightener-pulley. The tightener 
pulley is often inclined from the vertical, so 
that its projection is equal to the width of the 
driving pulley, in which case it not only serves 
to maintain a constant tension in the rope, but Pio. 13. 
it thus acts as a guide to conduct the rope from one groove 
to another. 


A modification of the usual belt-tightener is sometimes 
used for rope drives, as shown in Fig. 14, but this, it will 


be noticed, is positive in action and is only used to increase 
the tension on the ropes as the latter become extended 
with use. 

In tlie same way the tightener shown in Fig. 15 * is 
used to take up any slack that may occur in the rope. 

In this case the tightener-pulley T is mounted on 
a standard free to slide in the bottom guides G, A 
weighted lever L is connected to the pinion-shaft S by 
means of a ratchet-wheel and pawl (not shown in the 
figure). On the other end of this shaft and rigidly con- 
nected to it a pinion meshes with the rack R upon one of 

Fig. 15. — Dyblie's Rope-tightener for Dynamos. 

the guides and causes the standard and tightener-pulley 
to move along the base and thus automatically take up the 
slack as it occurs. A detent, D, in the carriage catches 
in the teeth of the rack and prevents the tightener from 
slipping back. It is evident that any form of adjuster 
which will not allow the tension-pulley to move in both 
directions — either back and forth or up and down — does 
not maintain a uniformity of tension ; as the humidity in 
the air may cause a rope to shrink very materially in a 
short time the tension on the rope will be greatly increased 
if the tightener-pulley is prevented from moving in.f 

♦Patented by J. A. Dyblie, Jan. 14, 1890. 

f Mr. Louis I. Seymour states that in a certain out-door drive at 



It is a well-recogiiized fact that atmospheric changes 
afteet the length of a rope, which in the presence of mois- 
ture always contracts. Experiments have shown that a 
dry hemp rope 25 feot long will shrink to 24 feet upon he- 


Pig. 16.— Tbnbion- 

I Variable Arm. 

ing wet.* It is for this reason that, in addition to taking 
up the slack caused by variation in load, provision should 

the Plymouth Cordage Company's Works in which li-inch rope is 
used to transmit 125 li. p. the tension carria^ is drawn in alK>ut 
eight feet by the shrinltage of the ropes during a severe storm ; that 
is, the rope is shortened about sixteen feet. The total length of rope 
in this case [s approximately 1600 feet, so that the cnntraction is thus 
about one per cent of its total length. The rope used was of 
Buperii>r quality maniila. laid with plumbago and tallow, otherwise 
the sUrinkage would liave been g-reatly in eic^ss of the amount 

• Indian EiigineeT, 1888. 


be made for maintaining a proper tension in the rope 
when the latter is variable in its length, due either to 
atmospheric changes, or permanent elongation as the rope 
loses its elasticity. This variation in length is particularly 
noticeable in rope-drives which are subjected to exposure 
from the weather. 

An arrangement sometimes adopted with horizontal car- 
riages as a substitute for pulley and hanging weight is 
shown in Fig. 16. In this arrangement the tail-rope, 
usually of wire, is wrapped around and secured to a 
grooved pulley-sheave keyed to a shaft which is fixed in 
position but free to rotate. A weighted lever is secured 
to the shaft bv means of a set-screw and maintains a ten- 
sion on the rope by virtue of its moment. It will readily 
be seen that this tension will not be constant, for the 
effective lever- arm of the weight, and hence the pull on 
the tail-rope, will vary with the position of the lever; thus 
in its normal position with the lever horizontal the tension 


in the tail-rope will be 7^= — ; biat if the tension-car- 
riage moves either in or out on its guides the lever will 
assume a new position as shown in dotted lines, in which 


case the tension will now be J' = . 


As the initial tension which gives adhesion to the slack 
side of the rope varies with the weight supported by the 
tension carriage, it is obvious that an increase of this 
weight will increase the power which may be delivered by 
the rope. As, however, the horse-power is proportional to 
the difference in stress in the driving and slack sides of 
the rope, the less weight on the tightener consistent with 
obtaining sufficient frictional resistance to slipping, the 
better will the ropes work. 

An example of rope-driving, in which tension-carriages 


are arranged to work vertically, is shown in Fig. 17, In 
this case the engine develops about 45 horse-power and 
runs at 90 revolutions per minnte, corresponding to which 
the velocity of rope is about 1700 feet per minnte. The 
fly-vheel Ib six feet in diameter, and is grooved for five li- 
inch 'Opes. The main sheave on the jack-shaf' is also 6 

Pio. 17.— ROPB-DHiyB wrni Vertical Tebbion-oarkiaobs. 

teot in diameter, and is grooved for five ropes. The rope 
passes continuously from the fly-wheel to the main sheave, 
making five wraps; then over the deflecting-sheave to tlie 
horizontal tension-carriage, and back to the fly-wheel. 

The jack- shaft sheaves are grooved for four 1-incti ropes, 
and are each provided with a friction-clutch, giving the 
line two shafts and, practically, the advantages of inde- 

-» ■ ^- •»! 



An example of such tranamission is given in Pig. 18. 
This drive transmits 150 h. p. for the main shaft at a to 
another at b, whose axis makes an acnte angle with the 
first, and which is several feet lower. The drive-sheave a 
is 5 feet 3 inches in diameter and runs at IGO revolutions 
per minute; it is grooved for six IJ-inch ropes, which have 
a velocity of 2800 feet per minute; c and d are idlers, the 
faces of which are nearly parallel to the drive-sheave; e 

Fig. 18.~-Rope-drive with Shafts 

and /are double idlers, there being two sheaves on each 
shaft, one 1^ inches less in diameter than the other. This 
arrangement of idlers in quarter- twist drives of this class 
has been introduced where there are more than four ropes 
in thb system, with the object of reducing the wear conse- 
quent to the friction produced by the side lead of the rope. 
In drives where there are more than eight ropes a cone 
has been used to a limited extent for the same purpose. 


Such a practice is, however, very unsatisfactory, as the 
trouble encountered by the differential driving exerted by 
each rope on a different diameter of the cone is greater 
than that which it attempts to obviate. However, by mak- 
ing the several steps on the cone separate, so that they 
form a series of independent idlers of different diameters,* 
the difficulty is overcome. 

It is evident that the employment of multiple idle 
sheaves of equal diameter possesses many advantages over 
a multiple-grooved pulley when used as guide-pulleys. 

In this case each sheave is independent of the others, 
and thus prevents in a large measure the evils due to dif- 
ferential driving and slip which would otherwise occur 
with fluctuations of load. 

With ordinary transmissions where the vertical distance 
between shafts is as great as 100 times the diameter of rope 
no trouble is experienced from the side lead of the rope, 
and, usually, no provision-is made to obviate it. 

An application of rope-driving to shafts at right 
angles, embodying several excellent features, is shown 
in Fig. 19. 

The plant is designed to transmit 250 horse-power from 
a 14-ft. fly-wheel, a, which is grooved for twelve 1-inch 
ropes. The line-shafts, m and n, are driven independently, 
and each drive has its own tension-carriage. The rope- 
sheave h is 72 inches in diameter, and is grooved for five 
ropes. At the side of the 72-inch sheave is a single- 
grooved idler, f, loose on the shaft, which serves as a guide 
for the rope to the tension-carriage. 

The substitution of a loose idler for an extra groove on the 
driven pulley in rope transmissions is due to Mr. Spencer 
Miller, although it has long been in use in cable-railway 

* This feature is the subject of a patent granted to Mr. John 
Gregg, March 11, 1890. 


practice; the advantage in its use is evident when we coa- 
aider that the tension -carriage, drawing out the stretch of 
the rope, mnst neceBsarily drag the first rope through the 

Pio. 19. — Transuission at Right Angles. 

groove of the pulley, which will require an excessive weight 
on the tightener pulley and a greater length of time before 


equilibrium is restored. By having this groove made into 
an individual wheel free to rotate on the shaft, this diffi- 
culty is overcome, and the transmission responds very 
freely to changes of load, so that when heavy machines 
are thrown on and off the ropes are not set in vibration, 
but the tension-carriage sheave K slides back and forth 
on the track, taking up the shock, with a minimum 
amount of wear on the ropes. 

In the present case the rope runs continuously around 
the fly-wheel and sheave from groove to groove. As it 
leaves the fly-wheel at the left hand it passes over the idler 
i to the tension -carriage sheave ^, which is suspended on 
adjustable hangers from a single-pipe track. This sheave 
is tilted by means of the adjustable hangers, so that the 
top is in line with the centre of the groove of the idler, 
arid the bottom is in line with the centre of the groove of 
the guide-sheave y, which serves to carry the rope back to 
the right-hand groove of the fly-wheel. 

The employment of multiple idlers instead of a multiple- 
grooved idle pulley is also shown in this figure, where power 
is transmitted from the shaft n to another o at right 
angles to the first. By engaging one or the other of the 
driven pulleys e or/ by means of a clutch, the shaft o may 
be driven in either direction. 

More recently ropes have been introduced to drive dyn- 
amos and special isolated machines, and where the dis- 
tance between dynamos and engine or driving-shaft is 
sufficiently great to allow a moderate sag in the ropes, such 
drives have been found to work very satisfactorily, provided 
other conditions are favorable. Where the distance be- 
tween shafts is limited, more wraps should be given to the 
rope in order to lessen the tension in each member. One 
great fault with dynamo drives is the use of too small a 
pulley on the armature-shaft. We can point to a score of 
plants using rope transmissions from a jack-shaft to dyn- 



Hino in wliich Die rope is overstrained and the djuamo 
jxillcy JH only half m large as it should be, in consequence 
of which the ropes are a constant source of trouble. 

With cotton ropes the i)ulley may be somewhat smaller 
than I hut nH(>(] for a similar size of nianilla ro|>e; but in 
iitiy cii.Mc there is a certain minimum diameter of pulley 
vvhiili Hhould ho used for any given rope. (See page 179.) 

Where the recpiired number of revolutions cannot be at- 
liiinccl with the Hize of jmlley imposed by the various con- 
ditiotiH, if tlu^ juitk-KJiaft cannot be speeded up nor a laro-er 
tlrivin^ -pulley uhihI, in such cases it would be better to take 
out the tope and put in a good leather belt. 

The hiinpleHt ainingement of rope transmission for djn- 
itnioM \H that in wliicli the rope is carried direct from the 
eii^'ine jly-wlieel to tlie grooved pulley on the armature 
himft, iiM Hhown iti Kig. 20, whicli represents the system 
iihcd in llm Htation of the Liverpool Overhead Railway.* 

There are four horizontal compound condensing engines 
with i'vlinderH LS.} and 'M inclies in diameter, 36 inches 
r^troke, nich of whicli is connected to a separate generator 
hy nieiihrt of I!) ropes IJ imdios diameier. Each engine is 
rated at 100 h. p. when running at 100 revolutions per 
minute with 120 pounds initial pressure; as the fly-wheels 
lue II feet in diauMrttM', the rope velocity will thus be about 
I loo feet per minuto. 

In ^^eneral it is more desirahle to drive the machines 
through an inlernuMJiate jack-shaft, especially so in those 
caHi'H when? a varying amount of current is required, as, 
for inHt,an(!e, in the lighting of public buildings. Such an 
arrangemc'iit, wIk^u the jack-shaft is provided with suitable 
fri(;t,ion or jaw clutches, will i)ermit machines to be thrown 
on or olT jih dc^sired. ^riie use of an intermediate shaft also 
perm its the attainment of the requisite speed of the dyn- 

• Fower, May. 1893. 


amo with moderate proportions of pulleys. In fact with 
many of the smaller engines in use a jack-shaft is essential 
if we wish to use rope-driving. 

Take, for example, a 75-h.-p. Corliss engine, running at 
85 revolutions per minute; the diameter of fly-wheel for 
this engine is 10 feet, and if we wish to drive the dynamo 
direct through a J-inch rope, the pulley on the armature- 
shaft should be, preferably, not less than 24 inches diam- 
eter; the speed of the armature would then be only 425 
revolutions per minute. To obtain a suitable speed, the 
driven pulley could not be more than about 12 inches in 
diameter, and with larger ropes this difference would be 
still more pronounced. 

A similarly rated high-speed engine runs at 230 revolu- 
- tions per minute, and is provided with a fly-wheel 5 feet in 
diameter; with the same 24-inch pulley on the dynamo the 
speed of the latter when driven direct would be not more 
than 570 revolutions per minute, so that in this case also 
the driven pulley would have to be reduced very much 
below that size which has been found best adapted to the 

It is true that the diameter of driving-wheel on the en- 
gine-shaft could be increased, and this is sometimes done. 
In the cases quoted the diameters necessary to give the re- 
quired speed would be about 20 feet for the Corliss and 7 
feet for the automatic engine. As these sizes give a cir- 
cumferential velocity within the limit of safety from the 
action of centrifugal force of the metal in the rim, it would 
be highly desirable to use such driving-wheels if other prac- 
tical considerations did not preclude their use. A driving- 
wheel 7 feet in diameter could readily be used on the high- 
speed engine without materially augmenting the journal 
friction, and the increased rim speed would be a beneficial 
factor in preventing momentary fluctuations due to change 
in load. 

4^ ROrE-l'lilVLXG. 

With the Corliss etij:ine, b.^wever. the increased weight 
due to A larire buildup dy-wheel "20 feet in diameter would 
usually debar its use on an en^.ne of this capacity; in ad- 
dition to this the large diameter would prevent its use in 
many KK»at ions, even if the iuoreiiseii weight and loss of 
|K^wer were no hiudraui.*e. Under these conditions the use 
of a jaok-shaft is the most suitable arrangement. 

In many cases it is desirable to use two engines so ar- 
rang\Hl that either or K>th may furnish the power to any 
one of several dynamos, jis shown in Fig. 21, which repre- 
sents a lTr>dK-p. n>jH^-transmission plant erected hy the 
Link-Wit Kiigiueering Company in the Virginia Hotel, 
Chioagi\ where two Corliss engines are each connected to 
a jack-shaft, having five counter-drives to the dynamos. 
Both the driven and driving sheaves on the jack-shaft are 
loose on tlie shaft, and are connected to it by means of fric- 
tion or jaw clutches, thus permitting either or both engines 
to be run, or any one of the five dvnamos to be thrown in or 
out of use. The positive jaw-clutch is used on the driven 
sheave, as it does not readily £:et out of order and is 
preferred by many engineers to the average friction- 
clutch, especially in those cases where much power is 

If it is desired to couple one engine to the shaft while 
the other is running, the former is speeded up until the 
loose driven sheave conies up to the speed of the shaft, 
when the dog-clutch may be readily thrown in gear with- 
out shock. With this arrangement there is a strong ten- 
dency for the bearings of the driven sheaves to heat when 
not coupled to the shaft; for this reason provision should 
be made to reduce the friction between the loose pulley and 
the shaft by relieving the tension on the rope when not in 
use, or, what is much better, the loose sheave should be 
mounted ui)on a hollow sleeve supported in pedestals inde- 
pendently of the shaft, as noted in description of plant 

If T, is the stress i 
the slack part, iind I'{=T,-'J\] 

a rope, 7", the stress in 
e the driving force, then, 


as we shall show subsequently^ the ratio of the maximum 

stress 7, to the driving force P, or -^, will vary from 

about 1.5 to 5.5 — depending upon the speed of rope, the 
coefficient of friction, and the angle embraced by the rope 
on the circumference of the pulley. In order, then, to 
have the transmitting force P as large as possible for a 
maximum tension T^ , the tension in the slack part of tho 
rope necessary for adhesion must be reduced to a mini- 
mum. To a certain extent this can be obtained by de- 
creasing the angle between the sides of the groove, but if 
carried too far this is a detriment rather than an advan- 
tage, for if the angle is sufficiently acute the rope will 
wedge and require more or less force to pull it out of the 
groove. The remaining expedient is to increase the arc of 

It can be shown that the friction of a cord or rope 
wrapped upon a fixed cylinder is independent of the di- 
ameter of the cylinder, and that it increases very rapidly 
with an increased arc of contact.* If the conditions are 
such that the coefficient of friction = one third, a ten- 
sion of one pound at the end T^ of the rope. Fig. 22, will 
support a strain at the end T^ of: 

1.69 pounds for an arc 

of contact 


lal to i coil. 



i - 



1 " 



2 " 

530 4'] • 


8 " 

4,348 ;. 5 


4 '* 

Therefore by increasing the number of wraps around the 
cylinder it is possible to increase the difference between the 
tensions in each part of the rope almost indefinitely. It 
will be noticed here that the rope is not in flying motion, 

Wi^W^^"^—— ^— ^— ■ - — - - -.— ■ ■■ — ■■■ ■■■!. ■■^■■1 ^1^11^ 

*Weisbacli, vol. i. p. 860. 


which would cause an equal centrifugal force to be set up 
in each member, thus altering the ratio of stress. As the 
centrifugal force varies with the square of the velocity, 
there is with an increasing speed of the rope a decreasing 
useful force and an increasing total tension oi? ihe slack 
side; but up to a given limit, which we shall subsequently 
show lies between 4000 and 6000 feet per minute, the 
total power transmitted for a given maximum tension in 
t-he rope will increase with the velocity. 

The great advantage thus obtained by increasing the ad- 
hesion was very early applied to numerous mechanical de- 
vices, chiefly for v. inding and hoisting purposes, and later 
for haulage systems. We are indebted to Willis* for the 
following quaint account and sketch (Fig* 23) of an ar- 
rangement for obtaining a continuous motion with a con- 
tinuous travelling-coil, first suggested by the author of the 
article, Sir Christopher Wren, over two hundred years ago. 


Read May 5, 1670. 

^* Having considered, that the ways hitherto used in all 
Engins for winding up Weights by Reaps have been but 
two, viz. the fixing one end of a roap upon a cylinder or 
Barril, and so winding up the whole coyle of roap ; the 
other by having a chain or a loose roap catching on teeth, 
as is usunl in clocks: but finding withall that both these 
• wayes were inconvenient the first, because of the riding of 
much roap in winding one turn upon another; the other, 
because of the wearing out of the chain or roap upon the 
teeth, I have, to prevent both these inconveniences, devised 
another to make the weight and its counterpoyse bind on 

* " Principles of Mechanism," p. 489 et seq. 


the cylinder, which it will doe if i6 be wound three tinitu 

"But because it will then in turning, scrue on like a 
worm, and will need a Cylinder of a very great length, 
therefore if there be two cylinders eaeh turned with three 
notches and the notches be placed alternately, the conyex 

edges to the concave as in the figure here adjoynod, the 
roap being wound three times about both cylinders, will 
bind iirmly without slyding and work up the weight witli 
a proportiouable counterpoyse at the other end of the 
The method of obtaining increased adhesion for a given 
y- ' """"""""""^^N number of coila in contact with 



each pulley has long been in use in rope-driving. In some 
of the earlier applications the grooves of the pulleys were 
semicircular in section, and of sufficient size to allow the 
rope to embrace the entire circumference of each pulley as 
represented in Fig. 24.* 

A better arrangement is that shown by Overman,f in 
which the advantage of the angular groove is obtained, 
and the ropes are not worn by rubbing against each other. 
This is shown in Fig. 25, and is thus described: 

£lec. World 

Fig. 25. 

"If the pulley A is grooved, of which at least two are 
fastened to the same shaft, the rope is directed on one of 
these pulleys, and passing around it goes to B, which re- 
volves on an inclined axis, such that the rope will be re- 
ceived from A' and delivered to A in the plane of the 
grooves. The number of pulleys may be multiplied to 
gain adhesion. This method of augmenting friction is 
preferable to the tension-roller, as no increase of tension 
is required; and it has the additional advantage of bend- 
ing the rope in the same direction, which makes it more 

* Willis. Principles of MechaDism. 
f Overman's " Mechanics," 1851. 



A similar arrangement was introduced in the San Fran- 
cisco Cable Railway in 1877. In this case. Fig. 26, the 
endless hauling-cable is passed alternately backward and 
forward over two grooved drums a sufficient number of 
times to obtain the necessary driving adhesion. 

Positive motion is imparted to the drum A, which hauls 
in the cable, while a second pulley, B, acting as an idler. 

Fig. 26. — Coil-prtctton, Street-railway Cablb. 

increases the frictional grip of the cable on the drum, by 
virtue of the increased arc of contact due to the number 
of wraps.* 

When, however, the cable permanently lengthens by 
stretching, the drum B may be moved further back by 
means of the sliding-base so as to take up the resulting 

By the use of this winder pulley the property of fric- 
tional adhesion produced by successive coiling is perfectly 

♦"Cable or Rope Traction." J. Bucknall Smith, C. E.; Etigv 
neering, London, 1887. 


effectual, for, although each coil is only in contact with a 
semi-circumference, the accumulation of frictional resist- 
ance is produced precisely as if entire circumferential 
grooves were employed. 

However desirable such winder pulleys may be for cable 
haulage or hoisting purposes, their advantage is greatly 
overestimated when applied to continuous rope- trans mis- 
sion. A little consideration will show that the frictional 
adhesion produced by a tension weight acting on a running 
rope with numerous wraps is entirely different from coil- 
friction. In the latter we have an accumulation of friction 
by which a small resistance applied at one end of the 
rope is able to hold an enormously greater load at the other 

In the continuous-rope system of power transmission, 
however, the load is distributed among all the wraps, so 
that when properly adjusted each wrap carries an equal 
proportion of the load and is subjected to an equal resist- 
ance on its slack side. There are few cases where the com- 
bination of the winder-pulley with the continuous- rope 
system offers any decided advantage over other methods. 

There is an incidental advantage in using a winder, 
especially in those cases where the difference in diameters 
between the driver and follower is quite appreciable; under 
such conditions the adhesion of the rope on each pulley 
may be made more nearly uniform by the employment of a 
winder, and there is less liability of the rope slipping in its 
groove, but this may usually be obtained more satisfactorily 
by other means (see page 168). 

In numerous cases ropes using winder-pulleys have been 
installed without regard to the work to be done or strain 
put upon the ropes, and many of the evils of rope-driving 
are directly traceable to this cause. 

Many engineers are opposed to using the winder-pulley 

50 K0PK-J)IUV1NG. 

in any form whatever, but occasionally it may be used to 

In outdoor or long-distance transmissions, or in special 
cases where it is desired to transmit a maximum power 
without undue stress in the rope, or in particular cases 
where the ordinary working stress may be exceeded, a 
winder-pulley may frequently be used to advantage, if the 
slack-side tension be reduced accordingly. Using a winder- 
pulley and increasing the back tension will permit a very 
large increase in the power transmitted; but since this im- 
poses an excessive strain on the rope, it soon wears out, and 
is a constant source of trouble. 

The gain in power by increasing the adhesion will be at 
the expense of journal-friction, which is thus augmented 
by the employment of wider-faced driving and driven pul- 
leys, in addition to that due to one or two more winder- 
pulleys; the wear of the rope, both external and internal^ 
will also, be greatly increased on account of the greater 
number of flexures given to the rope in passing over the 

The use of a winder-pulley at each end of a long drive 
in which only a single strand runs from the driver to driven 
pulley is an example of the application of coil-friction to 
the continuous-rope system; in this case both the working 
load and the back tension is carried by one rope instead of 
being distributed among several wraps, as usually happens 
in this system. 

That the percentage of gain is not as great as might be 
expected from the employment of coil-friction, will be seen 
from the following considerations: 

The ratio of tensions in the tight and slack sides of a 
rope running over two pulleys is dependent upon several 
factors, and may be determined from 

r, = r, (6*«), 


in which T^ = tension in tight side of rope, 
T^ = tension in slack side of rope, 
e = base of hyp. log = 2.7183, 
= coefficient of friction, 
a = arc of contact (circular measure). 

In this case the influence of centrifugal force is neg- 

Since the power transmitted by a wrapping connector 
is dependent upon the difference of tensions in the tight 
and slack sides, it is evident that with an assumed total 
tension T^ the available force for transmitting power will 
increase as T^ decreases. 

But T^ may diminish as €*» increases, that is, since e 
is a constant and <f> is constant for a given pulley and ma- 
terial, T^ decreases as a increases; hence if we increase 
the arc of contact the tension in the slack side of the pul- 
ley may be decreased in a ratio greater than unity, depend- 
ing upon the factors involved; in which case the net force 
P available for transmission will be increased, while the 
original assumed allowable tension remains the same. 

For example, if T^ = 200 pounds, <p = 0.3, a = 2.88 
{a° = 165°), we shall have 


and the net force P = 200 — 84 = 116 pounds. 

Under the same conditions, if we pass the rope from the 
driver over a winder-pulley back and forth twice, and then 
to the driven pulley and its winder in the same way, we 
should be able to transmit a little more than one and a half 
times as much power at the same speed without increasing 
the working tension in the rope. 


In this case a = 12.56 and -^ = 43.1; hence 


y, = 4.6 and P = 200 — 4.6 = 195 + pounds, 
llorse-power in second case _ 195 _ ^ 
' Horse-power in first case 116 

Now, if it were practicable to maintain the same slack- 
side tension jT, in these two instances and increase the 
driving-side tension under the conditions of the second 

case, viz. ~ = 43.1, we should have 

P=T,-T^ = (43.1 X 84) - 84 = 3536 pounds, 

and the ratio of power transmitted will now be , or 

thirty times as great as before. 

These results indicate that while we may vastly increase 
the driving-side tension for a given slack-side tension, by 
using a winder in tlie manner indicated, that is, with a 
single wrap connecting driver and follower, yet if we wish 
to maintain an assumed maximum working tension for a 
given-sized rope the percentage of gain will not be very 
great under the usual requirements of rope- driving. 

For a temporary drive the working strain may be in- 
creased to about twice the usual value, but for a permanent 
installation the usual working value should not bfe ex- 

Where a number of ropes are employed on a short-drive 
it is questionable whether the winder-pulley possesses suffi- 
cient advantages to warrant its employment in place of the 
continuous-wrap or individual-rope systems. In any case 
the conditions should be carefully considered, and the ac- 
tual gain compared with the various losses involved. 

A recent example illustrating the application of the 
winder-pulley is shown in Fig. 27, which represents the 
system of rope-driving installed by Messrs. Hoadley Bros, 
in the Fifty-second Street electric power-house of the 
Chicago City Railway Company. 


54 ROPK PBivmo. 

The plant is designed for 10 generators of the Westing- 
house A'o. 6 type, running 300 revolutions per minute. 
There are also to be ten 24-inch by 48-inch engines of the 
improved Wheelock type, arranged in five pairs, two ol 
wliich are now in operation. These run at 100 revolutions 
•per minute with 100 pounds boiler-pressure. The power 
usually varies from about 200 to 1000 h. p., but 'during the 
heavy traffic throughout the summer each pair of engines 
has frequently transmitted 1500 h. p. 

These engines have a built up fly-wheel (10 segments) 18 
feet in diameter, 39 inches face, which weighs about 
50,000 pounds. The rim is grooved for 21 wraps of IJ- 
inch man ilia rope. The driven pulleys are 6 feet in 
diameter, and contain 32 grooves for the rope, which runs 
about 5G00 feet per minute. Between the driven pulleys 
and the engine fly-wheel there is placed a C-foot winder, 
containing 11 grooves, around which the rope is car- 
ried before passing to the tension-sheave (Fig. 28), which 
in the present arrangement is placed horizontally above 
the engine near the ceiling, as shown in Fig. 29. Thus the 
rope is wound around the engine fly-wheel and the driven 
pulley, making 20 wraps; then it is carried' from the 
driven pulley to the winder back and forth 11 times, 
thence it is led over vertical guide-pulleys, 7 feet in diam- 
eter, to the horizontal tension-sheave 54 inches in diameter, 
then down over another vertical guide-pulley to the fly- 
wheel, where it started. By this means the arc of contact 
of each member of the driving-rope is increased practically 
180 degrees when all the ropes have adjusted themselves 
to the load, so that the power transmitted with the same 
tension in the rope will be about forty per cent more, if we 
neglect friction, than would be transmitted by the twenty- 
one wraps over the fly-wheel without the use of a winder. 

The net gain will be considerably less, owing to the vari- 
ous losses which this system entails. 




The application of ropes to transmit the power from a 
water-wheel to a line-shaft 40 or 50 feet or more above the 
axis of the wheel has latel}^ received considerable attention, 
and offers many advantages over the ordinary method in 
which a vertical shaft is nsed. The extreme weight of the 
latter in many cases makes it a difficult matter to provide 
a suitable bearing to support it. In such cases a horizon- 
tal turbine is used, and the wheel-shaft carrying the rope- 
pulley is extended and suitably supported, as shown in 
Fig. 30. 

The station of the Brush Electric Light and Power Co. 
at Niagara Falls is driven in this way. 

A line-shaft runs through the building, with one end ex- 
tending over the wheel-pit; to this are belted the gener- 
ators in the usual manner. Seventy-five feet below this 
shaft is located a 15-inch horizontal Victor turbine in a 
case of boiler-iron, its shaft extending to bearings supported 
by bridge-trees, which in turn are carried by the foundation 
I beams that support the wheel-case. This shaft carries an 
iron pulley 40 inches in diameter, grooved for 12 J-inch 
manila ropes. (Cotton was tried, but was not satisfactory 

The pulley on the driven shaft above is of wood, 70 
inches in diameter. The driving side of the ropes hang 
perpendicularly, and are free from the driver to the driven 

The slack side has two idlers or guide-pulleys, one of 
which is situated immediately below the driven pulley, and 
the other is about 20 feet above the driver. A tightener is 
adjusted in a running frame, in line with the driven and 
upper guide-pulleys. 

In putting on the rope the following course is taken : 
" Commencing at one side of the pulleys the rope is passed 
around from the driver to the driven pulley in every alter- 
nate groove until the opposite side is reached, thence 


around the tightener-pulley in the running frame, which is 
hung on an incline in such a manner that its discharg- 
ing side is in line with the side of the driver pulley 
whence we started . The remaining grooves are then filled, 
and the ends of the rope are spliced in position around the 
idler. Thus it is readily seen that there are two strands of 
the rope on the idler at all times. The object of this is to 
have one solid piece of rope on the idler at the same time 
that the splice is, so as to relieve the spliced piece of the 
strain of the idler. This system is giving far better satis- 
faction than has the upright shaft to those who have tried 

The plant designed by Mr. Kobert Cartwright for the 
electric station of the Citizens' Light and Power Company 
of Rochester, N. Y., is worthy of careful study, and may 
be considered a representative modern plant, adapted to 
use steam or water-power, and employing both ropes and 
belting, f 

Fig. 31 represents a cross-section of the station, and 
shows the general arrangement of the plant. The water- 
wheels are twin Poole-Leffel central-discharge turbines, 
23 inches in diameter, and at a speed of 560 revolutions 
per minute, under a head of 93 feet 6 inches, develop 500 
horse-power each, with a discharge of 3800 cubic feet of 
water per minute. The wheels proper are made of phos- 
phor-bronze, with backets of Otis steel, tinned. The wheel 
bed-plates are heavy cast-iron box sections, machined and 
bolted together with heavy bolts fitting reamed holes. The 
wheel-shaft is 4J inches diameter, running in adjustable 
babbitt-lined bearings. A rope-wheel 4 feet in diameter is 
keyed on the shaft, and is grooved for fifteen IJ-inch 
manilla " Stevedore '' ropes, made with four strands and a 
core, worked in with plumbago in the process of making. 

— ■ 

* F. E. Pritchard, Elect. World, April 16. 1892. 
t See Trans. Am. Soc. C. E., vol. xxx., 1894. 




From the 4-ft. wheel 15 ropes run to a rope-wheel on the 
line-shaft above, 76.8 inches in diameter, and grooved for 
sixteen IJ-iDch ropes. The ropes being endless, the idler- 
strand is passed over a 5-ft. single-grooved wheel, placed 
in a movable frame. The frame traverses in iron guides 
and maintains by its weight a constant tension on all the 
ropes. This is made adjustable for the amount of tension 

Fig. 82.— Plan of Power Statios. 

by the application of counter- weights to the frame. The 
speed of the line-shafts is 350 revohitions per minnte, and 
the rope travel is 7037 feet per minute. The water-wheeis 
are supplied from a steel flume 7 feet in diameter. From 
the horizontal portion of the flume a 4-ft. pipe leads down 
to each wheel, and has a geared 48-inch Chapman valve at 
the lower end, between pipe and penstock, as shown in 



Pig. 31. These valves are fitted with a 12-iiich by-pass, 
for the purpose of equalizing the pressure on both sides of 
the large valve in opening or closing. 

A horizontal " Woodbury ^' compound condensing slide- 
valve engine, with extra heavy bed-plate, is set in the 
power-room at point marked in Fig. 33 " Engine N^o. 1." 
Steam-cylinders are placed with the large cylinder out- 
side, so that pistons and rod may be easily removed. Cylin- 
ders are 19 inches and 31 by 34 inch stroke. At 167 revolu- 
tions, with a boiler-pressure of 110 pounds per square inch, 
vacuum 32 to 24 inches and cutting off at y^^ stroke, the 
engine is rated at 500 horse-power, and is guaranteed to 
produce a horse-power on an evaporation of 30 pounds of 
water per hour. The crank-shaft is a steel forging in one 
piece. Journals are 11^ inches diameter by 21 inches 
long. Crank-pin 11| inches diameter by 8 J inches long. 
The end carrying the rope-driving wheel has an outboard 
bearing. Governor balance-wheel is 8^ feet diameter by 
25-inch face. Rope-driving wheel is cast in halves 10 feet 
6 inches diameter, and grooved for fifteen If -inch ropes. 
These ropes lead to a 5-ft. rope-wheel on the line-shaft 
above, with same arrangement for tightener as is applied 
to the water-wheels. Rope speed of engine-drive is 5500 
feet per minute. 

The line-shafts are of hammered iron 5 inches in diameter, 
and arranged with heavy floor pedestals, fitted with self- 
adjusting, ring-oiled, babbitt-lined bearings. The rope- 
wheels are placed on heavy cast-iron quills, furnished with 
Hill friction-clutches of 500-horse power capacity each. 
By a series of jaw-clutches, pulleys, and belts any line- 
shaft can be operated from any water-wheel or engine, all 
the line-shafts making the same number of revolutions. 
Fig. 33 shows in detail the quill, clutch, and bearings. 



TuJ^i use of ropes in connection with portable tools and 
trave) ling-cranes has long been established, and their con- 
venience and adaptability to a wide range of work make 
them a necessity in many shops. The advent of the small 
electric motor in our machine-shops will, however, proba- 
bly replace to a large extent all other forms of special 
transmission for portable tools, as it is already replacing 
countershafts and belting for machine-driving in many 

One of the greatest fields of usefulness for rope-driving 
is in the transmission of power to a moderate distance, 
under conditions which are unfavorable to the use of belts 
or shafting. 

With rope-driving one is enabled at a comparatively 
small cost to transmit power in any direction to a building 
remotely situated from the source of power, which would 
otherwise require a long and expensive line-shaft or an in- 
dependent engine or other motor. The facility with which 
it may be carried in any direction across rivers, canals, and 
streets, above or under ground, up hill and down, over 
houses and into buildings, is a feature very favorable to 
the further extension of rope transmission; but the rapid 
progress which has been made in the development of elec- 
trical transmission has limited the economical application 
of ropes to moderate distances. There are, however, cer- 
tain limits between which the transmission of power by 
ropes is yet more efficient than by any other known method. 

The employment of ropes for this purpose — i.e., trans- 
mission of power to a distance — is not a recent application, 


The first method of transmission of power to any consider- 
able distance was made in 1850 by C. F. Hirn at Logel- 
bacb, near Colmar, Alsace.* " The works consisted of a 
large number of buildings separated at some distance from 
one another, which were required to be changed into a 
weaving factory. As there was but one steam-engine on 
the works, the expense of transmitting power to the various 
buildings by ordinary shafting (the shortest length of which 
was 84 yards), or of erecting separate engines, would neces- 
sarily have been great; and the desire to obviate this ex- 
pense resulted in the adoption of the telodynamic system. 

The first plan adopted was the use of a band of steel 172 
yards long, ^ inch thick, and 2 inches broad. This was 
slung as an endless band over two wooden rollers or 
pulleys, 6 feet 6 inches diameter, which were placed 84 
yards apart and made 120 revolutions per minute, giving a 
speed of 28 miles an hour in the band. In practice this 
plan was found to be open to two objections : the lightest 
wind agitated the banJ, and the pulley-guides tore it at the 
points of riveting, whilst the guides themselves were 
rapidly worn out. Notwithstanding these objections this 
plan rendered valuable service, and continued in operation 
for a year and a half, transmitting 12 horse-power to one 
hundred looms. 

"The difficulties of the flat band suggested round wire 
ropes i inch diameter ; these were accordingly substituted 
and were placed upon the same wooden pulleys, which, 
however, were first grooved to the depth of half an inch.f 

This plan answered every expectation, and experience 
having fully sanctioned its use a second wire rope was soon 
put in operation, transmitting the power to a distance of 

* Mr. H. M. Morrison, Proc. Inst, M. E. 1874, p. 57. 

f Prof. Unwin, in his Howard lectures (see Electrician, Feb. 3, 
1893), states that an English engineer, Mr. Tregoning, suggested the 
Bubstitutioa of the wire rope. 


about 770 feet. The two pulleys were each 9 feet 6 inches 
diameter, making 91^ revolutions per minute, and a steel 
rope i inch diameter was employed transmitting 50 h. p. 
at a speed of 31 miles per hour. In this instance it was 
found necessary to have supporting pulleys to prevent the 
rope from trailing upon the ground. These carrying pul- 
leys were placed half-way between the transmitting pulleys, 
or 128 yards apart, and in the first instance they occasioned 
very great difficulties by the rapidity with which they were 
worn out in the groove. They were constructed succes- 
sively of copper, wood, and polished cast iron, and were also 
faced with leather, horn, india-rubber, lignum-vitae, and 
boxwood. All these failed, however; the facings were soon 
worn out, and when the groove was of metal or hard wood 
and did not itself wear it destroyed the rope. After re- 
peated experiments a dovetailed groove was formed in the 
bottom of the pulley groove and filled with gutta-percha, 
(as shown in Fig. 65, page 186.) 

" This turned out a perfect success, and carrying pulleys 
thus faced have an almost unlimited amount of durability." 
Fibrous ropes were used in the United States for long dis- 
tance transmissions a few years later ; thus we find a com- 
munication in the Scientific American* in which a cor- 
respondent from Winsted, Conn., speaks of several rope- 
drives in his vicinity, one of which had been in use since 
1858. "It transmits the power for a manufactory, em- 
ploying several circular saws, across the river, 225 feet dis- 
tant, by a I -inch rope running over two pulleys six feet in 
diameter, at a speed of 5600 per minute. The pulleys are 
sheltered, but the rope runs exposed in all kinds of 
weather, needing no attention except at times to be rubbed 
with grease having a very small amount of rosin mixed 
with it." 

♦Vol. Ill, 1861, p. 215. 


Ufa Pia 

66 ftoPE-DRiviKa. 

When power is taken from a water-wheel in locations 
where land is not available for buildings, the use of 
ropes as a means of transmitting the power from the 
wheel to the mill or factory forms a most economical ar- 
rangement if the drive is properly designed for the work. 
It is a great advantage in many other cases to have the 
power plant and the several buildings of a works isolated 
from each other; this is especially desirable in sawmills 
and wood -working establishments, where the risk from fire 
is greatly reduced by such an arrangement. 

An example of rope-driving in which the conditions are 
particularly adapted to this form of transmission was 
erected a few years ago at Portland, Ore. The mill is built 
on piles situated in the Willamette River, while the engine 
and boiler room are upon the solid ground, some distance 
away. The engines are a pair of Wheelocks, with cylinders 
33 by 60 inches, intended to be speeded to 70 revolutions 
per minute. The fly-wheel is 24 feet in diameter by 66 
inches face; it is built up of ten segments grooved for 
thirty-three IJ-inch manilla ropes, and fitted to a shaft 20 
inches diameter; its weight is 40 tons. 

Two ropes are taken from this wheel to jack-shafts 35 
and 45 feet distant; the driven pulleys — one with 16 and 
the other with 17 wraps — are each 76 inches in diameter, 
and are keyed to 10-inch shafts. From the end of eacli 
jack-shaft a 600-h.-p. transmission is arranged and 
carried to the mill. Each shaft is fitted with a friction 
clutch to allow either of the mill transmissions to be thrown 
out if desired. 

The general arrangement, showing location of driving 
and driven sheaves, is represented in Figs. 34 and 35, from 
which it is seen that power is delivered to two shafts 7 
inches in diameter — in the one case at a diagonal distance 
of about 200 feet, and in the other at a distance of 185 feet 
from their respective drivers — both the driven shafts being 


at right angles with the jack-shaft. Each drive was 
designed to transmit 600 h. p. with a rope velocity of 7550 
feet per minate, but it has since been found advantageous 
to reduce this speed to about 6000 feet per minute. 

The arrangement of ropes in these transmissions is 
similar to that used in the Chicago City Kail way Company, 
illustrated on page 53. In the present case only three wraps 
of l^inch rope are used to convey the power from the jack- 
shaft to the mill, but in order to prevent slip and decrease 
the tension in the slack part of the rope the driving and 
driven pulleys have each nine grooves, six turns being 
carried around another pulley or winder, thereby increas- 
ing the arc of contact and, hence, the adhesion. In this 
plant the stress in the rope is very much greater than that 
ordinarily used. 

It is evident that the whole strain must be borne by the 
three strands, as it is only the difference in tension of the 
tight and slack sides of the ropes that can be used to trans- 


mit power; since = h. p., we find the difference in 

. . p 3300Q X 600 __^^ , - ,, 

tension, F = ^^^-^ = 3300 pounds : and as there 


are three wraps, the difference in the stresses in the two por- 
tions of the rope will be 1100 pounds. As we shall find 
later the total stress will be greater than this, due both to 
the action of centrifugal force in the rope and to the force 
necessary for adhesion. 

As the maximum working tension for a 1^-inch rope is 
usually only about 450 pounds, it is evident that each 
wrap carries nearly three times its proper load, taking the 
wear and life of the rope into account. The engine has 
only developed about 700 horse-po jr as yet, so the total 
stress in each rope has been very much less than the 
above, — probably not more than 300 to 350 horse-power on 


each drive; bnt even with this reduced Btrese one rope was 
repla^^ed inside of fifteen monthe. 

For short distances shiifting is often employed for such 
transmissions, but with this latter the friction of the jour- 
nal-bearings is a very important consideration, and effectu- 
ally debars its nse for long-distance transmission. 

This will be seen from the following considerations: 

Jjet 6 = distortion of shaft (circular measure) per unit 
length ; 
(f = distortion in degrees; 
/ = unit length of shaft; 
L = length of shaft in feet; 

r = distance of outer fibres from axis = ^ ; 

d = diameter of shaft; 
PR = twisting moment on the shaft; 
N = revolutions of shaft per minute; 
V = velocity of circumference of shaft = wrfJV; 
= modulus of torsion of the material 

= two fifths of the modulus of elasticity; 
/ = maximum torsional stress in the outer fibres 

~" mV ' 
W = weight of shaft = 3.36 pounds per foot per square 

inch of section; 
F = load due to friction. 

-mi n fi 32PRI 

If the angle of torsion ie giveo in degrees, then 

„ «• X3» 

'' = 'mr 

of length will be 

therefore the angular distortion per foot 


27r Gr nO d ^ ' 

The working limit of the angle of torsion for steel shaft- 
ing ought not to exceed 0.10 degree per foot in length of 
the shaft; that is, 0^ = O.IOZ, and for wrought-iron shaft- 
ing 6^° = 0.075Z;* assuming the shaft to be of steel and 
Substituting the corresponding value for 6° in (3), we obtain 

O.IOZ X TtGd = 360/ X 12L; 

hence /= 800d if we assume tliat G = 11,000,000 pounds. 
Since the horse-power transmitted by the shaft equals 

-P-^X ^K7:?^9 if we substitute the value of PB, (= -r^Y), 

ooOOU \ lo / 

there is obtained h.p. = -—;-/' "' ^^^ ; but the velocity at 

^ 16 •' 33000 •' 

the circumference of the shaft is v = ndN, also/= S00d\ 


h.p. = O.OO^bd'v (3) 

If the bearing is well worn and fitted to its shaft the 
resistance due to friction will probably lie between the 

limits ^0TF and -(f)Wy\ or between 1.570 Tf and 1.28 

(t>W^ where is a coefficient, which in the present case we 

shall assume equal to 0.06. 

Taking the lesser value, we shall have ^= — 0Tr, 

where F is the force at the circumference of shaft : eces- 
sary to overcome the journal friction. If there are no 

pulleys on the shaft W= -j^L X 3.36; the horse-power 

exerted to overcome the friction will then be 

Fv 4^ Ttd'Lx^Mv ^^^^^y 
^•P- = 3-300-0=^^><4 33000 = OO'^^^- ^x- 

* Reuleaux : Der Konstrukteur. 
f Unwin. 


pressed as a ratio, the percentage of power required to 
overcome friction will be 

h,p^ _ O.O'Sd'Lv 
h.p. ~ O.0O95d'tf' 
from whicli there ia obtained 

■ 1585' 

• (4) 

That is, for a steel shaft whose diameter is one inch the 
horse-power recjiiired to overcome the friction in a length 
of 1583 feet will be equal to the total allowable transmit- 
Ciug capacity of the shaft. 

For wrougiit iron ^^ = .0751 and h.p. = .007 5d'v, from 
which may be determined the value of the ratio 

^_:P:i-nnnns-^- nr hr. ^ .^ x — P-. 


= 0.0008-j 



The following tables, calculated from these formulee, 
give the limits at wliich the power transmitted hy a shaft 
is absorbed by the friction of the bearings, the assumption 
being that the factor of journal- friction equals 0.06, and 
that the allowable twist shall not exceed 0.10 degree per 
foot of length for steel, nor .075 degree for wrought iron. 
The shaft is supposed to be free from all pulleys and gears. 

Tahle I.— Limit op Lenoth fob Steel Sbaftiko. 
A'o puUeyt on the tine. 


I^npth in feet 


Lenrth o'hea 
















Table II.— Limit of Length for Wrought-iron Shafting. 

N^a pulleys on the line. 

Diameter of 

shaft in 


Length in feet 

when total 

power is absorbed. 

Length when 
efficiency = 
50 per cent. 

Length when 
efficiency = 
75 per cent. 









From the foregoing it will be seen that shafting is alto- 
gether unsuitable for conveying power any considerable 
distance, and as belting is not adapted to this work choice 
must be made of some other method. 

For a mere dead pull, such as the alternate strokes 
needed to operate a pump, work is, and has long been, 
transmitted to very great distances; as by the long lines of 
draw-rods, ropes, or wires used in mining regions, quarries, 
and elsewhere, for transmitting the power of a water-wheel 
by means of a crank on its main axis, pulling during half 
its revolution, against a heavy weight at the end of the 
line, and thus storing up energy for the return stroke. 

Wooden pump-rods were used in this manner about 1865 
near Petroleum, W. Va. A large condensing engine was 
located in a central position, and the rods transmitted the 
power to a number of oil-wells, twenty-seven in all, situated 
at various distances and in different directions from the 
source of power. The greatest distance was about four 

Posts with crank-arms were used to change the direction 
of the pull. The rods were of hickory, connected end to 
end by means of iron straps. 

The transmission of power from the famous 72-foot di- 
ameter overshot wheel at Laxey, on the Isle of Man, is by 
means of similarly connected trussed rods, which in this 


case are supported at regular intervals on small wheels 
ranning on iron ways. About 150 horse-power are trans- 
mitted in this way. 

This method was adopted on a very large scale in the 
mines of Devonaliire for the transmission of power from 
large overshot water-wheels to pumps fixed in the shaft of 
the mine at a considerable distance higher up the valley. 

In one case* the water-wheel was 52 feet diameter, 12 
feet breast, and its ordinary worlting speed was 5 revolu- 
tions per minute. The length of stroke given by the crank 
to the horizontal or " flat " rods was 8 feet; the rods were 
3i-inch round iron, and were carried on cast-iron pulleys. 

At Devon Great Consols, near Tavistock, there are alto- 

Piei. 86.— RoPK-DRiVHS WTFH Bknt Crahks AT 130 Dbbrbbs. 

gether very nearly three miles of 3-inch wrought-iron rods, 
carried on bobs, pulleys, and stands, whereby power for 
pumping and winding is conveyed along the surface to dif- 
ferent parts of these extensive mines, from 11 large water- 
wheels ranging up to 50 feet in diameter, to which the 
water is brought along eight miles of leats 18 feet in width. 

Rods and wire ropes have also been used to transmit ro- 
tary motion to a considerable distance in a similar manner 
by pliiulng the cviiuks at 130 degrees, as shown in Fig. 36. 

It is evident t.liat the distance of transmission by this 
contrivance will be subject to the sag of the ropes, unless 

•■■TheO!d Whenl FrieDdsbip Mine, near Mary tavy," Proc, Inst. 
M. E, 1881, p, 100. 


intermediate shafts are employed. The motion must also 
be comparatively slow, owing to the severe strains which 
would be thrown upon the bearings and pins by the surg- 
ing and swaying of the ropes during the rapid changes of 
motion to which they would be liable. In order, then, to 
transmit much power, heavy rods or large ropes would be 
necessary, and under these conditions economical transmis- 
sion would be limited to short distances. 

Among the various means in use at the present time for 
conveying. power to a distance we find steam, water, gas, 
compressed air, electricity, and rope systems. Each of 
these has its own applications and advantages, but it must 
be borne in mind that with the exception of rope trans- 
mission, of which numerous examples have already been 
given, all other forms usually require a generator at the 
one end and a motor, with separate attendants, at the other. 

Other things being equal, the relative merit of various 
methods of transmitting power will be indicated by the 
cost of transmitting a certain amount of power to any 
given point, as compared with the cost of this power at 
the generating station, while their absolute merit will be 
shown by comparing the cost of the transmitted power at 
the receiving station, with the cost of producing the re- 
quired power directly at this point.* Such determinations 
are materially affected by variations in the amounts of 
power and in the distance of transmission; the other prin- 
cipal factors to be considered being the efficiency of the 
system, the number of working hours per annum, the 
price of 1 h. p. per hour at the generating and receiving 
stations, and the convenience and applicability of the sys- 
tem to each special case. 

The efficiency of any system of transmitting power is 
expressed by the ratio of the power obtained at the receiv- 

*Stahl, ** Wire-rope TransmissioD." 


ing station to the power given out at the generating sta- 
tion. In all systems losses of power of greater or less 
magnitude occur^ and the most efficient system is that in 
which those losses are reduced to a minimum. We shall 
not attempt here to lay down rules governing the choice of 
any particular method, for the requirements and condi- 
tions are so varied that every individual case must be 
decided upon by the engineer separately with a knowledge 
of all the facts before him. Our present object is to ascer- 
tain the principles governing the use of ropes, and to 
determine those conditions best suited to their economic 



The subject of rope-driving may properly be placed 
under two heads, according to the nature of the material 
composing the ropes, whether metallic or non-metallic. 
With few exceptions metallic or wire ropes are used almost 
exclusively on long-distance or telo-dyn;uiiic transmission, 
while non-metallic ropes are employed for intermediate and 
comparatively short drives, the consideration of which con- 
stitutes the present subject-matter.* 

Among the materials employed in this method of power 
transmission we find special forms of leather belting used 
as ropes working in V grooves; fibrous ropes, including flax, 
hemp, cotton, and manilla, are, however, chiefly employed. 

Rawhide ropes, which are made from f inch to 2 inches 
in diameter, are used to a limited extent. Where the 
stress in the rope is not great and the accompanying slip 
is small, rawhide works very well, and will last from three 
to six, and in some cases ten, years. Under ordinary cir- 
cumstances it is not necessary to use any dressing, as suf- 
ficient lubrication is furnished by the rope itself; if the 
rope slips in its groove the leather will be burned, and lose 
its flexibility, and also its adhesive qualities, to a certain 
extent. A rawhide rope has very little tendency to rotate 

* Concerning wire-rope transmission the reader is referred to the 

"Wire-rope Transmission" (A. W. Stahl); ''Elektrischen Krafts 
tlbertiagung" (A. Beringer); "Drahtseiltriebs" (D. H. Ziegler); 
'* Constructeur" (F. Reulcaux); "Machine Design" and "Central 
Stations" (W- C. Uuwin). Also, trade pamphlets published by W, 
^, Roebling & Softs; Cooper, Hewitt & Co.; and others. 


on its axis; for this reason the wear is not always uniform, 
and with a heavy tension it is liable to take the set of the 
groove in which it runs. This is rather an advantage for a 
straight drive, where the rope always runs in the same di- 
rection; but in those cases where a rope is led on to the 
pulleys at an angle this will be a disadvantage, as under 
such conditions the rope often slips, and wear is excessive. 
Where the rope is subject to wet or dampness, rawhide is 
an excellent material to use, as it is very little affected by 
dampness. The cost of rawhide rope will average about' 
six times that of a good quality of manilla transmission- 
rope, and although it is to be preferred in certain cases, its 
greater cost will limit its application. 

Round-leather ropes, formed by twisting narrow strips of 
leather into a continuous spiral, are used for light driving, 
and are very desirable for some classes of work. 

Solid round-leather ropes, made from several thicknesses 
of belting cemented together and secured with screwed 
wire forced into the leather, are made in various sizes up 
to 2| inches in diameter, but sizes larger than f or 1 inch 
in diameter are seldom used. 

Steel ropes with leather washers closely threaded on 
have been tried with considerable success, but the expense 
of such a rope would necessarily limit its application. 

Other special forms of leather belting used as ropes are 
found in the various modifications of the square and angu- 
lar belts which have been used for a number of years for 
both light and heavy drives. 

Leather ropes as large as 1} inches square, made up of 
layers of leather cemented together so that the whole is 
uniform and continuous, have been used to replace quar- 
ter-turn flat belts, and also for main driving. 

These run in V grooves so that the adhesion is 
greatly in excess of that produced by a flat belt on a 
smooth pulley under the same tension. In the same 


way triangular belts built up from various thicknesses of 
leather possess the advantages characteristic to all forms 
of rope-driving which use a V groove, viz., greater ad- 
hesion for a given tension, and the facilit}' with which such 
transmitters lend themselves to the communication of 
power between shafts at an angle with each other. 

It is evident that several of these leather-rope belts may 
be used side by side in a manner similar to the various ap- 
plications of fibrous ropes. Such ropes have proven satis- 
factory in those cases where the pulleys are of approxi- 
mately the same diameter; but on pulleys whose diame- 
ters vary considerably each portion of the leather rope in 
contact with the driver tends to rotate the follower at a 
different velocity, necessarily producing slip and wear, to 
an extent depending upon the ratio of the diameters em- 

In England manilla is now being used very largely, but 
cotton ropes were formerly^ preferred to the exclusion of all 
others for all kinds of driving; but the most probable cause 
of this was not that cotton was the best or most economi- 
cal material for the purpose, but that rope-driving is most 
common at cotton factories, and cotton ropes were made 
in the locality by men who were familiar with the local 
product, and had been for years making spindle and rim 
bands of small size. When the demand for large sizes 
arose these rope-makers applied themselves to the newer 
industry, and shut out other materials.* . 

In the mills of Dundee and vicinity, and in the North of 
Ireland, where flax and hemp are worked, we find ropes of 
hemp, a local product, used entirely. 

In many cases ropes of cotton are to be preferred, as they 
are generally softer and more pliable than the ordinary 
manilla ropes, thus allowing smaller pulleys to be used 

* W. H. Booth, Am. MacJiinist,* January, 1891. 


with less injury to the fibres. In fact, cotton ropes of 
small diameter have been used for years in cotton ma- 
chinery bandings over pulleys, and under conditions which 
would wear out a manilla rope in one third the time. 
There is also an advantage in that there is less internal 
chafing and wear when the rope is bent over a pulley, on 
account of the smoothness of the fibres and the great elas- 
ticity of the yarns. 

The cotton fibre is not, as it appears to the eye, a solid 
cylindrical, gossamerlike hair, but when fully ripe is shown 
under the microscope as a fiattened hollow ribbon or col- 
lapsed cylindric tube twisted 
several times throughout its 
length, as shown in Fig. 37;* 
it is of equal size for about 
three fourths of its length, and 
it then gradually tapers to a 
point. This point is a section 
almost perfectly cylindric, and, 
unlike the rest of the fibre, 
often composed of solid matter. 

Covering the outside mem- _ _„ ^ ' ^_ ^ 

, 7,1 /?! . 1 . Fig. 37,— -Cotton Fibre, Or- 

brane of the fibre is an oleagi- leans Variety {Oossypium 

nous coating generally known Hirsutum). 
as cotton-wax. This wax amounts to about two per cenl 
of the fibre, and in the spinning of the material it requires 
to be reduced to a certain point of liquefaction by the 
heated temperature of the room before it can be made to 
work properly without lapping on the drawing rollers. 

These fibres vary in size from 0.00084 inch mean di- 
ameter, and about ^ inch long to 0.000635 inch mean 
diameter and 2^ inches in length, depending upon the 
variety of the cotton; but for a given variety the differ- 

* See "The Cotton Fibre," by Hugh Monle, Jr. Published by 
Haywood & Son, Manchester, Eng. 


ence is very small: thus in the Sea Island cotton the 
maximum length of fibre is 2 inches, while the minimum 
is If inches; in the same way in the Orleans variety 
shown in Fig. 37 the maximum length is 1^ inches and 
the minimum || inch. 

As a rule, those cottons which have the longest fibres 
are also the smallest in diameter: they possess the natural 
twist in a more perfect and highly developed form, and 
are much stronger and more elastic. 

In all good commercial fibres of cotton there is neces- 
sarily (1) a very small percentage of solidified oleaginous 
matter distributed over the internal surface of the fibre 
deposited when the vital fluids were in active circulation; 
and (2) a certain percentage of moisture known as water 
of hydration. 

These together with the twisted structure impart to the 
fibres that suppleness, tenacity, and elasticity without which 
they would be almost useless for manufacturing purposes. 

The cotton fibre is thus naturally well adapted to the 
work of being twisted into yarns; the presence of the 
natural convolutions and comparative smoothness of the 
surface of the unit filament permits considerable elonga- 
tion, and the wax on its surface serves as a natural lubri- 
cant and prevents the fibres from becoming brittle. 

Thus it will be seen that ropes made from fibres pos- 
sessing these characteristics are particularly well adapted 
to the transmission of power in which the rope is con- 
stantly undergoing a varying strain and is subjected to 
much flexion. 

The strength of cotton ropes is, however, relatively small 
when compared with other fibrous ropes, and although the 
weight is about one fifth less than mauilia, for equal di- 
ameters, the actual first cost is from fifty to seventy-five 
per cent greater than for the latter. 

Nystrom gives the breaking strength of three-strand 


cotton ropes at less than one tenth that of similar man ilia 
rope, but this is apparently too low for a good quality of 
transmission rope. 

Tests made at Watertown on a number of Lambeth 
ropes varying in size from 1 inch to 2| inches in diameter 
indicate that the breaking strength is equal to about 
4000c?' pounds, while the extension varies from twenty to 
twenty-five per cent, corresponding to a reduction in 
diameter of about fifteen per cent. The weight of these 
ropes is very closely 0.26d* pounds per foot of length. 

A series of tests carried out by Kircaldy * on cotton 
ropes ranging in size from ^f inch to 2y\- inches in di- 
ameter give the breaking strength as 3700c?' pounds for 
minimum value and 58006?' pounds as a maximum. The 
weight per foot of these ropes varied from 0.25c?' to 0.29e?' ; 
the extension under a stress of about 85 per cent of the 
breaking load varied from 17 to 27 per cent. 

Reduced to a common basis in which the strength is 
made proportional to the weight, and averaging the results, 
we find that the breaking strength may be represented by 
4600c?' pounds. 

The data on cotton ropes are too meagre to determine 
whether their strength decreases as the diameter increases, 
but this is probably the case. 

Reuleaux gives 7500 pounds per square inch of section 
for cotton transmission -ropes, which agrees very closely 
with the above values. 

From the formula, breaking strength, S = 4600c?' pounds 
the values given in Table III have been calculated, and 
may be considered as representing approximately the 
strength of cotton transmission -ropes of good quality. 

The working strength of cotton transmission-rope may 
be taken higher, in proportion to its ultimate strength. 

*See also Kent's " Mechanical Engineer's Pocket-Book," 


Table III. — Strength of Cotton Transmission-ropes. 

iSpJ^J^^^^^^^ UltimateBreaklng^Strengrth, 

i " 1,150 

f 1,800 

f 2,600 

1 3,500 

1 4,600 

li 7,200 

H 10,400 

If 14,000 

2 ia,4oo 

than is used with manilla, for the latter is weakened by 
the grease with whioh it is lubricated; and, moreover, a 
larger factor must be allowed for wear on account of the 
character of the manilla fibre, which breaks more easily 
under bending strains. 

As compared with manilla, then, the advantages of cot- 
ton ropes of the sanafe ' diameter are: Greater flexibility, 
greater elasticity, less internal wear and loss of power due 
to bending the fibres, and the use of smaller pulleys for a 
given diameter of rope. Its disadvantages are: Greater 
first cost, lesser strength, and, possibly, a greater loss of 
power due to pulling the ungreased rope out of the groove 
— in any case this is usually small with speeds over 2000 
feet per liiinute. 

As we have already noted, manilla rope is used very ex- 
tensively for transmission purposes, but its application has 
not always met with that success which would follow u 
more thorough knowledge of its requirements. Inefficient 
rope-drives are erected and run for a few months, or per- 
haps only days, and. are replaced with larger ropes if the 
sheaves will permit, or, as in many cases, tiie ropes give 
way to leather belting, and henceforth rope-driving is con- 
demned. The true cause is not so much the inefficiency 
of the ropes as it is the lack of knowledge governing their 


use and application; in order to obtain a proper concep- 
tion of this a study of the striieture of the rope will be 
found advantageous. Manilla, or, more prop- 
erly, nmnilla hemp (abaca) rope, is made from 
the fibres of the Afusa textili>i, a plant closely 
allied to the banana, growing near Manilla, 
in the Philippine Islands. The fibres are a 
part of the outer covering of the leaf-stalk, 
which attains a length gometinies as great as 
15 feet. To obtain fibres of suitable size for 
manufacturing rope the leaf stalks are sub- 
divided, and in the process of segregation 
the fibre assumes an appearance somewhat 
similar to that produced in^plitting a piece 
of wood — it is rough and uneven, and more i 
or less splintery throughout its length. Those , 
fibres, although in themselves not very large, 
are composed of very fine and much elongated 
bast-cells, which overlap each other as shown 
in Fig. 38. The cells are irregular in outline 
and vary considerably in size. The length 
of the cells is about one fourth of an inch. 
A series of tests on nianilla fibre carried out 
by Dr. Stanley M. Coulter of Purdue Univer- 
sity, shows that the cells are not, as com- 
monly supposed, held together by an inter- 
cellular tissue or mucilaginous substance. A 
cross-section of a portion of the fibre, (a). 
Fig. 39, enlarged 450 times, shows that there 
are no Intcicelhiliir spaces; and various reac- 
tions to determine the presence of i 
or other vegetable glue revealed no traces of 
its presence. 

The characteristic roughness possessed by the manilla 
fibre is due entirely to mechanical causes, sncb as, for 

Fro. 38. 
■ Cells of Ma. 
[LLA Fibre, 



kistance, the laceration of a cell in the separation from the 
leaf-stalk, or the subsequent opening out of the ends of 
the cells. 

The contour of the perimeter is also rough, as noted in 
the figure^ 39fi, as it retains the form impressed upon it by 
the contiguous cells when in the plant. 

These fibres have great strength in the direction of their 
length, but are weak transversely;* when made into rope 
thej are compelled, in bending over the elieave, to slide on 



each other while under pressure from the load. This 
causes the internal chafing and grinding which, if not pre- 
vented, soon wears out a rope when subjected to bending 

In addition to the action of the fibres upon each other, 
the strands and the yarns of which the strands are com- 
posed also slide a small distance upon each other, causing 
friction, and hence. internal wear. 

By opening out an old dry rope which has been used over 
a sheave, a fine powder will be disclosed, showing that 
where the rope was bent over the sheaves the strands, in 
sliding on each other, ground some of the fibres to powder. 
Another reason for this is that the fibres in an old rope be- 
come brittle and weak when dry, so that the constant 

* The teiiBile strength of n 
mnda per sqaare inch of sec 

will average over 30,004! 


flexure to which they are subjected rapidly disintegrates 
the cell congeries. 

Aside from the external wear which a rope suffers from 
contact with its sheave (part of which is the differential 
driving effect), these two are the principal causes of rapid 
wear in a rope-drive, to remedy which we must in the first 
case lubricate the fibres, and in the second, prevent undue 
flexure of the rope. How this is effected in practice will be 
seen presently. 

BOP£-DBiyiira. 85 


Ik manufacturing manilla rope the fibres are first spun 
into a yarn, this yarn being twisted in a direction called 
right-hand. From 20 to 80 of these yarns,* depending 
on the size of the rope, are then put together and twisted 
in the opposite direction, or left-hand, into a strand. 
Three of these strands for a 3-strand, or four for a 4-strand, 
rope are then twisted together, the twist being again in 
the right-hand direction. It will be noticed that when the 
strand is twisted it untwists each of the threads, and when 
the three strands are twisted together into rope it untwists 
the strands, but again twists up the threads. It is this 
opposite twist that keeps the rope in its proper form. 
When a weight is hung on the end of a rope the tendency 
is for the rope to untwist and become longer. In untwist- 
ing the rope it will twist the threads up, and the weight 
will revolve until the strain of the untwisting strands just 
equals the strain of the threads being twisted tighter. In 
making a rope it is impossible to make these strains exactly 
balance each other, and it is this fact that makes it neces- 
sary to take out the turns in a new rope — that is, untwist 
it when it is put at work. 

* A three-strand rope one inch in diameter is the key to the sizing 
of the yarns. Yarns of 20s are of such a size as to require 20 to fill 
a tube half an inch in diameter, or to make one strand of an inch 
rope; 26s requires 26 to fiU the same size tube ; and so on. The size 
of cotton yarns, on the other hand, depends upon the number of 
hanks per pound ; thus 20' s, or number 29 cotton, requires 20 hanks 
(840 yards each) of this size to weigh one pound. 


The fibres of maniila which are thus twisted into ropes 
will average over 6 feet in length, varying from 3^ to Vi 
feet. If they were long enough, the most advantageous 
method of nsiiig them would be to lay the fibres side by 
side, aud secure them at the two ends; each fibre would 
then bear its own share of the strain, and the strength of 
the bundle would be that of the sum of the strengths 
of the separate fibres. As a long rope could not be 
formed in this way, the fibres are secured by twisting 
so as to produce sufficient compression to prevent them 
from moving upon each other when a strain is applied ; but 
in attaining this amount of compression their strength is 
greatly reduced; this very compression acts as a constant 
weight on the fibre, and must be deducted therefrom before 
the available strength can be applied. 

The weakening effect produced by twistingvaries consid- 
erably among the fibre,i of the same rope according to their 
distance from the centre or heart of the bundle. If a cer- 
tain Amount of twist be given to a bundle of fibres, the 
outer ones will be strained more, and will act with leas use- 
ful effect than those on the inside, which will have to bear 
the greater part of the strain while the rope is being used. 
It will therefore be evident that if the fibres were twisted 
at once into a thick rope the outer fibres would be so much 
strained as to be of little or no use in coutributing to the 
strength of the rope, but by making the rope as we have 
indicated by first twisting into yarns, then into strands, and 
finally combining these into a rope, the strain is more 
equalized, and the important properties of length and 
strength are secured without too great a sacrifice of the 
strength of the individual fibres. 

The degree of twht in the rope may be determined by 
constructing a right-angled triangle, the base of which is 
the circumference, mitl the height the length of one turn 
of the strand measured parallel to the axis. The difference 


between this height and the hypothenuse is the quantity by 
which the rope is twisted. The ropemaker's ordinary rule 
for a three- strand rope is to have one turn to as many 
inches as are contained in the circumference of the rope; 
but the degree of twist is variable and more or less de- 
pendent upon the judgment of the maker. 

Experiments by Keaumur to determine the effect of twist 
upon a rope showed that a small well-made hemp cord 
broke in different places with a mean weight of 65 pounds; 
while the three strands of which it was composed bore 29^, 
33^, and 35 pounds respectively, so that the total absolute 
strength of the strands was 98 pounds, although the aver- 
age real strength was only 65 pounds, thus showing a loss 
of 33 per cent. 

More recently the test of a small rope showed an average 
strength of 4550 pounds, while the aggregate strength of 
its 72 yarns was 6480 pounds — each yarn bearing about 90 
pounds; thus there is a loss of 1930 pounds, or about 30 
per cent. 

Ropes made of the same hemp and the same weight per 
foot, but twisted respectively to two thirds, three fourths, 
and four fifths of the lengths of their component yarns, 
supported the following weights in two experiments made 
by Duhamel:* 

Pounds. Pounds. 

Twothirds 4,098 4.250 

Three fourths 4,850 6,753 

Fourfifths 6,205 7,397 

The results of these experiments led Duhamel to make 
ropes without twist by placing the yarns together and 
wrapping them round to keep them together. The rope 
had great strength and pliability, but not much durability 

* TomUnson, vol. vii. p. 574. 


on account of the outer covering wearing away or opening 
when bent, thus admitting moisture to the interior, which 
rotted the yarns. 

In general, the greater the twist the more hard and rigid 
the rope is, and the better it will keep its form; but it is 
not as strong, weight for weight, as the more loosely 
twisted rope; moreover, the hard -twisted rope is more rigid 
than the other, and is not as suitable for transmission pur- 
poses, owing to the rapid wear which constant flexure 

A very excellent transmission-rope, known as the " Lam- 
beth,'^ is made in a somewhat similar manner to that just 
described, but it is not open to the same objections. In 
this case the rope, which is of cotton, is made of four 
strands twisted together in the usual manner, but the 
strands are themselves composed of a bundle of fine yarns 
and have scarcely an appreciable twist. Each bundle, 
comprising many hundred yarns, is wound spirally with 
smaller bundles of about 100 yarns each. In this way 
the outside of the rope acts as a shield or covering to 
the cores which do the work. By this means a certain 
amount of the natural elasticity of the cotton is re- 
tained, and its pliability is much greater than in ordinary 
hawser-laid ropes. 

Lubrication of transmission-ropes is provided for in 
various ways. Frequently the rope is laid up dry and a 
coating or dressing is given to the exterior, which is sup- 
posed to penetrate to the interior and lubricate the fibres. 
With some dressings this may occur, but with others 
the effect is merely local; the interior ot the rope re- 
mains dry, and much bending soon wears it out. With 
cotton ropes, as we have already noted, the internal 
chafing and wear is very much reduced, and for this 
reason cotton ropes are laid up dry and are not usually 


lubricated; they are, however, generally coated with some 
form of dressing to prevent the fibre from rising or the 
rope fraying, and to protect it from an undue amount of 
moisture when exposed to the weather; it also assists in 
retaining the natural moisture in the fibres, without 
which they would become brittle and weak. 

According to the nature of the dressing, the interior 
may or may not be affected by the outer coating. Bees- 
wax and black lead, with a little tallow, forms an excellent 
moisture-proof covering for ropes; it fills in the spaces 
between the strands, and the rope soon assumes a perfectly 
round and smooth appearance, like a bar of iron; with this 
coating the interior retains its natural condition, and should 
therefore not be used where a lubricant is desired. Pine 
tar is much used on cotton ropes for the same purpose. 
Mixtures of tallow and black lead; molasses and black 
lead; equal parts of resin or beeswax, with black lead, 
tallow, and molasses melted together, and applied hot; and 
various other compounds are in use for this purpose. 
Tallow, lard, and other greases are used separately, and are 
fairly satisfactory as a lubricant. When the rope runs out- 
of-doors a water-proof coating is necessary to preserve it 
from decay, and for this purpose, if it is also desired to 
lubricate the rope at the same time, there is probably noth- 
ing better than tallow and black lead, or graphite, pro- 
vided the rope is not twisted too hard, which would pre- 
vent the dope from penetrating. In certain drives that 
are subjected to hard service and are more or less exposed 
to the weather boiled linseed -oil is used very successfully. 
According to Mr. A. D. Pentz,* the rope is treated every 
two weeks to about two quarts of the oil, dripped one drop 
at a time upon one of the sheaves, which is uncovered on 
top; the rope runs on the bottom of the sheave and slowly 

* Eng. Magazine, Nov. 1893, p. 250. 


absorbs the oil. Although the rope is very materially 
weakened by this process, yet the greater freedom of the 
fibres permits a heavier working strain to be carried, for it 
is the relative wear of the fibres that determines the life of 
the rope. A manilla rope with the fibres properly lubri- 
cated will, under the same conditions, outlast from two to 
four similar dry-laid ropes which are allowed to run dry. 
Manilla transmission-ropes are generally laid up in tallow 
paraffine, soapstone, or a mixture similar to the above 

A superior manilla transmission-rope is that known as 
the " Stevedore,^' or black rope, which is made with both 
three and four strands, the latter being laid up about a 
central core. The yarns of this rope are each coated with 
a mixture of graphite and tallow, so that when twisted 
into strands the coating lodges in the hollows and uneven 
places among the fibres, and thoroughly lubricates the 
strands and individual fibres composing the rope, which is 
thus made practically as nearly water-proof as possible. 
Alter it has been in use a short time its appearance is that 
of a black rod of iron, smooth and round, similar to the 
beeswax-coated rope previously mentioned, but perfectly 
flexible. A manilla rope thus made will last from three 
to eight years if not overstrained ; when running indoors 
under favorable conditions the latter limit may be attained^ 
but when exposed to the weather, or when working under 
less favorable conditions, its life will be shortened. 

Hemp ropes intended for outdoor service are sometimes 
treated by passing the yarns through boiling-hot tar, suit- 
able machinery being used to regulate the amount of tar 
retained in the yarns so that the fibres may be coated over 
and thus preserved from decay. Tarring protects rope 
from injury by exposure* to rain and immersion in water, 
but it makes its fibre rigid and impairs its strength; for 


this reason it is unsuitable for ordinary transmission pur- 

It has been shown by experiment: * 

1. That white or untarred rope in continual service is 
one third more durable than tarred. 2. That it retains its 
strength much longer when kept In stock. 3. That it re- 
sists the ordinary injuries of the' weather one fourth longer. 

With the exception of the outside yarns of large haw- 
sers, manilla ropes are not tarred. 

The breaking strength of a rope depends both upon 
the quality of the material and the degree of twist given 
to the strands;* for a loosely twisted rope of a given diame- 
ter the strength is less than that in a hard twist of 
the same diameter, but compared weight for weight, the 
rope with the lesser degree of twist is the stronger. 

In discussing the strength of ropes, which formerly was 
always given in terms of the circumference, there is a lack 
of uniformity among writers in the relation between the 
diameter and area of a rope. The circumference, as meas- 
ured by a tape, depends upon the number of strands in the 
rope and their compression upon one another. If the 
strands still retained their circular section when twisted 
into a rope, the circumference of a 3-strand rope would be 
2.86 times the diameter of the circumscribing circle, as 
given by Nystrom; if the strands completely fitted the 
circle its measure would be n times the diameter as given 
by I/nwin and others. 

As neither of these conditions obtains in practice the 
true value lies between 2.86 and 3.14, and we shall assume 
3 as the most suitable factor. 

In the same way the area of the cross-section of the rope 
is variously estimated as that of the area of the circular 
strands, or as the area of the full diameter of the rope. 

*Duhamel ; Traite de la fabrique des maiiGenvrcs pour Ics vaisscaux. 


By aBsumiDg the rope to be made of three strands, the 
cross-section of each of which is a circle, the area of the 

rope would evidently be 3(^5'), where 6 is the diameter 

of each strand. If d represents the diameter of rope, i.e., 
the diameter of circnmscribing circle, the area in terms of 
d will become 

The section of the strands, taken at right angles to the 
axis of the rope, is, however, not a circle, lis can be seen 
from Fig. 40. The degree of twist given to the strands, 

Fig. 40.— Acre; 

and the compression of the latter npon one another, will 
evidently affect the area of the section; for the longer the 
spiral the more nearly will the cross-section of each strand 
approach a circle. It is obvious that the true value must 
lie between 0.52(r and O.'TSMd'. In determining this area 
the writer made a number of plaster casts at different 


points of several S-stmnd manilla ropes, varying in size 
from J inch to IJ inches diameter. With these casts as 
dies, which were covered with printer's inli, impressions 
were made and the area obtained by using a plauimeter, 
Tlie area at several sections was found to be practically 
constant for each rope, and varied between 0.61rf' and 
0.65(P — the mean value being O.QSd', from which we 
obtain the ratio 

El of section of rope 

Area of circumscribing circle 0.7854(7' * ' 

that is, the actual area of a 3-strand rope equals eight 
tenths of the area of the circumscribing circle. 

Fia 41 —Cross-section 

A print obtained with a plaster cast, showing the distor- 
tion of the strands from a true circle, is reproduced in 
Fig. 40. Fig, 41 is a similar print, showing the relation 
between the actual area of the cross-section and the area 
included between circular strands and the tangents joining 
them. It will be uoted that the e-tcesa of area outside of 


the lines drawn tangent to the inner circles is but slightly 
greater than that between the tangents and the inner 

An inspection and tabulation of the results of numerous 
tests on manilla rope * shows that the strength per square 
inch increases as the diameter of the rope decreases. 
Formulas for the strength of rope based upon the circum- 
ference and a constant multiplier — as, for instance, S = 
800c% where S is the breaking strength and c the circum- 
ference — must be regarded as giving only an average value 
for diameters approximating those experimented upon. 
As the strength of good manilla rope varies from 10,000 
pounds per square inch for a 2-inch rope to over 12,000 
pounds for a half-inch rope, it can be seen that a more 
accurate determination may be made if a variable is used 
in the formula. With the above assumption of area ratios, 
obtained by trial, the following expression has been de- 
duced, which will give a very fair value of the breaking 
strength for new manilla ropes: aS^^ = lOOcrx, in which x 
is a variable depending upon the diameter of rope; for 
manilla rope we may assume the empirical value 
a: = 81 — 9d, where d is the diameter in inches. 

The ultimate strength per square inch of actual section 
will then be 

0.8 ^d' 

From these formulas Table IV has been computed : 
The above values are for new manilla ropes made of 
selected stock; ropes that are greasy or wet will be reduced 

* Major Parker's report of tests made at Watertown Arsenal, 
1885. Ex. Doc, No. 36. Experiments of M. Doboul in ••Bulletin 
de la Societe d'Encoiivagement des Arts/' Paris, 1888. Riehle Bros,' 
laboratory tests ; and otheii. 



Table IV.— Strength of Manilla Transmission-ropes. 

Dia. of rope = d. Breaking Strength, strength per square inch, fif. 

i 1.900 12.200 

2.900 11,950 

4.100 11,750 

5,500 11.525 

1 7,100 11,350 
H 8.800 11,125 
li 10,900 10,975 
li 15.000 10,625 
If 19,800 10,300 

2 25,100 10,000 

in strength from 20 to 30 per cent, but when ropes are 
used for transmission of power the lubrication of the 
fibres is of more importance than the actual breaking 
strength of the rope, as in any case the apparent working 
strain, as calculated from the power transmitted and the 
speed of the rope, should not be taken greater than 5 per 
cent of the ultimate strength; in many cases not more 
than 2 per cent is used. 

If we wished to obtain the working strength in 
any given case we would usually divide the breaking 
strength by an assumed apparent factor of safety, but 
for a flying rope, in addition to this, it is necessary 
to provide a factor of wear; moreover, the actual strains 
are generally much greater than the normal as calcu- 
lated from the power transmitted, due to vibrations in 
running, imperfect tension mechanisms, defects in con- 
struction, and other causes. As the strength at the splice 
is only 70 to 75 per cent of the strength of the rope, the 
actual margin for wear and unknown strains is not as large 
as would at first seem apparent. In the first place, tlifc 
strength of a lubricated rope is weakened about 25 p^ 
cent by the grease, then 25 per cent more must be deduct" ^ 
for the strength of the splice; this leaves only about^-2 
per cent of the original strength of the rope whic 



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available for trail sraission of power. To allow for possible 
imperfections in the rope, due to its manufacture or the 
material used, we must allow a further reduction of, say, 
one fifth (that is, 10 per cent of the original strength of an 
unlubricated, unspliced rope); thus we have only 40 per 
cent of the original breaking strength, which we may con- 
sider as the actual working strength. Allowing 10 per 
of this as the apparent working strain (equal 4 per cent of 
original strength), we obtain 36 per cent as the actual 
margin for wear and unknown stresses which may be set 
up in the rope; that is, we have practically only an actual 
factor of safety of ten, used in its ordinary acceptation, 
instead of twenty-five, as would appear from the low work- 
ing stress. 

The working strain in executed rope transmissions will 
be found to vary considerably, as shown in Table V ; 
but plants which have been successful, as well as 
those in which the wear of the rope was destructive, indi- 
cate that 2000?" pounds is an economical working strain.* 

As this value is such a small percentage of the breaking 
strength, it is unnecessary to use a coefficient varying with 
the diameter of rope, as tlie difference is not worth con- 
sidering. From data furnished by the Messrs. Pearce 
Brothers, of Dundee, who have erected rope-belting exten- 
sively. Prof. TJnwin shows that in different cases in practice 
tlie driving force, or difference in tension, on the two por- 
tions of the rope is equal to 75 to 80 d^ pounds; that is, 

r, - T; = P = 75 to 80 d\ 

where T^ = tension on driving side; 
T^ = tension on slack side; 
F = driving force. 

* See paper by C. W. Hunt, Trans. A. S. M E., vol. xn. 
page 230. 

110PE-DRIVIN(3^. 99 


It is probable that the value of the ratio -~ varies from 

1| to 2^ in ordinary practice, depending upon the speed of 
the rope, the coefficient of friction, and the arc of contact 
between rope and pulley. 


TJnwin assumes -^ = 1.2 when the belt embraces 0.4 of 

the circumference of the smaller pulley; hence the greatest 
tension would be 

7; = 1.2P = 90tZ*to96rf*. 

This is based upon a coeflBcient of friction = 0.7 for a 45° 
groove, which we believe to be greatly in excess of its aver- 
age value for running ropes. Moreover, the rope velocity 
was not considered in determining the above ratio; at 
speeds over 2000 feet per minute the influence of cen- 
trifugal force cannot be neglected, as it produces a very 
considerable force in the rope, and for these reasons the 


value of the ratio -~ will be greater in average practice 

than 1.2. 

If we assume a speed of 4000 feet per minute, the value 
of this ratio for greasy ropes may be taken equal to 2, from 
which there is obtained T; = 2 X 75^' to 80fl?' = IbOd^ to 
160c?' — a value somewhat less than that which we have as- 
sumed as a suitable working strain, viz., 200^" pounds. In 
a recent communication from Messrs. Combe, Barbour & 
Combe of Belfast, who have been engaged in furnishing 
rope transmission for thirty years, it is stated that their 
basis of calculating the horse-power is to assume that a 
rope 6^ inches circumference (if inches diameter) work- 
ing on a four-foot pulley going 100 revolutions per minute 
will drive 8 h. p. under medium circumstances; and by 
"medium circumstances*^ they mean '* that the ropes must 
work at a distance of at least 20 feet from centre of shafts 




and at a less inclination than 40° from the horizontal, at a 
speed not under 2000 feet per minute. Should the ropes be 
working vertically or at an angle greater than 45°, instead 
of taking 8 h. p. as the basis, you should take 7 or 6 respec- 
tively, according as the conditions grow worse and worse. 
On the other hand, should the ropes work horizontally 
and at a greater distance than 20 feet from centre to centre, 
and with a speed up to 3600 feet per minute, and the pul- 
leys be fairly large, say from 5 to 7 feet in diameter, you 
may take 10 instead of 8 as the basis. *^ From these consid- 
erations the working strain may readily be determined. If, 

as before, we take — * = 2, we find that under average con- 
ditions the allowable working strain will be T^ = I3bd^ 
pounds. Under more favorable conditions and a higher 

velocity -^ will be greater and 7\ will approach 200t?\ 

Although we have assumed the normal working load not 
to exceed 200^' pounds, this must be considered as the 
economical load for the lasting qualities of the rope; in 
many cases, however, the first cost, the more convenient 
adaptation of smaller ropes, and the use and lesser cost of 
smaller pulleys outweigh the greater economy obtained by 
the larger ropes, and loads are carried far in excess of that 
given. By the use of a greater number of wraps the work- 
ing load on each will be reduced; but this adds to the first 
cost of the plant, and many concerns prefer to put in a new 
rope every year or two rather than put in more or larger 
ropes every six or eight years. 

The ropes most commonly found in use vary from f inch 
to 2 inches in diameter, although other sizes are frequently 
employed ; in one case, cited by Mr. T. S. Miller,* a rope 

* Trans. A. S. M. E. 1891. 


no larger than ^ inch diameter is used very satisfactorily 
to transmit 20 h. p. 

The largest rope in use for this purpose, so far as the 
writer is aware, is 3^ inches in diameter; it is of cotton 
(Lambeth), and is used to drive the cable drums in the 
Washington Street power station of the North Chicago 
Street Kailway Company. 

In England and Germany ropes smaller than 1 inch 
in diameter are seldom employed except for machine 
driving ; in ordinary cases the sizes most frequently 
adopted vary from 1^ to 2 inches in diameter. 

In main drives where a number of ropes are used, the 
tendency, as judged from recent practice, seems to favor a 
diameter about 1§ or If inches. 

With a given velocity and working tension the weight of 
rope required for transmitting a given horse-power will be 
the same, irrespective of the diameter of rope; the 
smaller rope will require more parts, but the weight will 
be the same. As we have stated, many engineers prefer 
to use a greater number of ropes over wide-faced pulleys; 
but in order to reduce the expense incident to a large 
number of grooves in the pulleys a closer margin is allowed 
on the smaller ropes, and in consequence the normal work- 
ing strain on each rope is usually increased far in excess of 
that which would be necessary if the same weight of rope 
were employed as for the larger diameter. As a result of 
this, the small ropes ai'c rapidly worn out, and frequent 
renewals become necessary. 

We are aware that small ropes are advocated and in- 
stalled by many engineers, but we believe this to be wrong 
both in principle and practice. In rope-driving the first 
cost and erection of small driving-pulleys may influence a 
designer to use small ropes; but as ropes are sold by the 
pound, and as the weight necessary to transmit a given 
power should be the same, irrespective of the diameter, it 

102 BOPE-DBlTINa. 

is evident that the first cost of the rope itself will be the 
same, whether large or small ropes are employed. More- 
over, if the pulley is proportioned to the size of rope in 
each case, the smaller rope will last only about one third as 
long as a rope twice its size, under similar conditions. 



However desirable it may be to use a given diameter of 
rope, the conditions of the problem frequently prohibit the 
employment of such ropes, and the designer must deter- 
mine whether he shall use a smaller diameter or a different 
material or method of transmission. 

If we assume that there is a minimum diameter of pulley 
which may be safely used for any given diameter and speed 
of rope, it will be evident that the number of revolutions 
of the pulley imposes conditions which limit the choice of 
rope diameter. 

Thus if the maximum speed of rope be taken at 5000 
feet per minute for a permanent installation, in which the 
working load is 200d^ pounds, and the least diameter of 

pulley D = c?i-^ (Vv) + 1^" (page 179), then the greatest 
number of revolutions which can be obtained under these 
conditions with a 1-inch manilla rope will be 550. 

If the least diameter of pulley for cotton ropes be taken 
equal to 0.8Z>, then the greatest speed will be approxi- 
mately 700 revolutions per minute for the same diameter 
of rope. If a greater rotative speed be desired, it is evident 
that a smaller rope must be used. 

In any case, the greatest number of revolutions which 
may be obtained without excessive wear for a given diame- 
ter of rope will be found in Table VI, which has been de- 

termined from the formula iV"= — =:, in which 

N = revolutions per minute of smaller pulley; 
V = velocity of rope in feet per minute; 
D = least permissible diameter of pulley* 



For machine-driving greater speeds are obtained by the 
use of smaller ropes than those given in the table; but it 
is not advisable in most cases to use a rope less than f inch 
diameter for general transmissions. 

Table VI. — Greatest Revolutions per Minute for given 

Diameter op Rope. 




Maximum Revolutions per Minute of Smaller Pulley 

corresponding to a Linear Velocity of 

6000 Feet per Minute. 









The wear of a rope is both internal and external. As we 
have previously noted, the internal wear is due to the bend- 
ing of the fibres and their sliding upon one another, which 
produces a grinding action, — very much increased when the 
strands are not lubricated or when a hard twist is given to 
the rope, thus preventing by the greater compression of the 
fibres upon one another that freedom of action which is so 
esseatial to the life of the rope. It is evident that a similar 
compression of the fibres will occur when a rope under ten- 
sion is wrapped around a pulley: the greater this tension 
the greater also will be the compression and distortion of 
the fibres. 

The external wear is due to the contact between the rope 
and the sides of the groove in which it runs, and is greatly 
increased when slip occurs; roughness in the groove also 
increases the wear, and for this reason the rim should be 
turn^ smooth and polished, as the outer fibres, rubbing 


Oil a rough-turned or cast surface, will gradually break, 
fibre by fibre, and thus give the rope a short life. 

Contact between diffej'ent ropes, or between a rope and 
some obstructing surface, such as a partition wall, post, or 
floor-beam, is frequently the cause of a large portion of the 
external wear of a rope : this may be due to faulty construc- 
tion or erection; pulleys designed with too small a pitch 
between the grooves, or running out of true, causing the 
ropes to vibrate and flap against each other, or, as in out- 
side work, a swaying, producing contact, may be set up in 
the ropes, due to the action of the wind. 

Excessive swaying will also tend to cause the rope to 
jump its groove. In order to prevent this and reduce the 
side motion as much as possible it is often customary in 
outdoor drives to place idlers for both tight and slack 
sides of the ropes so as to guide each portion as it enters 
upon or leaves the groove. 

A characteristic uniform surging sometimes occurs in 
flying ropes due to the harmonic vibration which is set up 
when the speed and distance between shafts bears a cer- 
tain relation to the tension in the ropes. Cases of exces- 
sive vibration due to this cause have been remedied by 
slightly increasing or decreasing the speed of the ropes. 
Where such vibration causes the rope to beat against an 
obstruction, as a floor or ceiling, the external wear is of 
course increased. 

With the same total stress in a rope, it may be assumed 
that the wear, both internal and external, increases directly 
with the number of flexures, the slip, and the surface in 
contact; and also, that a reverse bending is more injurious 
to the rope than single bends in a constant direction. 
For a given speed the number of flexures and the actual 
surface in contact with the pulleys will decrease as the 
distance between centres increases, and hence the wear 
will vary inversely with the distance between centres of 


pnlleys, but it must be noted that witb imperfect con- 
struction an increAsed distance between shafts will favor 
swaying and nibbing of the ropes against each other and 
the edges of the groovea. 

The number of flexures and the surface in contact will 
evidently increase directly as the velocity, and therefore 
the wear may be assumed to vary directly as the velocity of 
the rope. If we assume that two li-inch ropes are neces- 
sary to transmit a given horse-power, it will require eight 
f-inch ropes to transmit the same pewer at the same speed 
aiiil tentiion per square inch of section. If suitable pul- 
leys tire used in each case the wear will be considerably 
greater with tlie smaller ropes. For the total external 
surface of the eight ropes in contact with the pulley each 
revolution is twice as great as that obtained with the IJ- 
iuch ropes; moreover, the distance between centres of 
shafts will generally be considerably less with the smaller 
ropes, and as the number of revolutions of the smaller 
pulleys should be more than twice as great for the same 
speed of rope (since the pulleys are less thiin half the she 
of those used for the larger rope; see page 180), the number 
of flexures and the wear of the rope on tlie pulleys will be 
greater with the |-iucii rope ; for not only is each rope 
bent more than twice as many times per minute, thus pro- 
ducing eight times the bending in the smaller ropes, but, 
IIS the slip is independent of the diameter of the rope, it 
will be evident for the same proportional stress and arc of 
2ont:ict that the siip will be four times greater in the case 
of the smaller ropes, even if we neglect the differential 
driving effect which may be assumed to increase with the 
number of ropes. 

Tims it will be seen that under similar conditions and 
proportional stress, we should expoct the smaller rope to 
wear out more than twice us fiist as one double its size; 
and when the stress is proportionally greater in the smaller 


rope, as we ordinarily find it, the wear will be still 

These conclusions are borne out in practice, for in trans- 
missions using small ropes, J-inch in diameter and under, 
the life of manilla ropes is usually only from six to twelve 
months; in many cases such ropes will last only three 
months, although others have been in active service for 
periods varying from one to two years. 

On the other hand, the larger-sized ropes, one inch to 
two inches in diameter, will last from two to six years, and 
under favorable conditions large ropes have lasted eight 
and even ten years. 

The same is true regarding cotton ropes. 

The comparatively short life of small ropes used in 
machine-driving has led to the belief that cotton ropes 
wear out rapidly; but such an impression, at least regard- 
ing the larger sizes, is altogether erroneous, as these ropes 
when properly put on and cared for will give good service 
for ten or twelve years. 

Messrs. John Musgrave & Sons, Bolton, Eng., who have 
had a large experience with rope-driving, state that some 
of their ropes have been in use for seventeen years and were 
still in good order. 

The life of a rope, whether of manilla or cotton, will de- 
pend altogether upon the work it has to do and the atten- 
tion it receives. 

Two ropes cut from the same coil can be put to work on 
different drives, and one will last only six months while 
the other will be in good condition after continual service 
for ten years. 

From an investigation of numerous examples of rope- 
driving under a variety of existing conditions the writer 
is led to believe that ropes less than f inch diameter 
., should not be used if it is at all practicable to employ the 
larger sizes, and that ropes one inch in diameter and over 


are to be preferred where it is possible to use the larger 
pulleys which are necessary for such ropes. 

Witli larger ropes the wear is not only much less, but, 
where the nsual multiple-wrap system is used, when a 
number of yarns give way the rope does not part at once 
if subjected to a greater or sudden tension, but may run 
until a convenient opportunity offers to shut down; where- 
as with the small rope, having a greater number of wraps, 
when a strand or a number of yarns give way, any in- 
creased stress due to additional load or imperfections in 
the system is liable to still further rupture the yarns in the 
weaker rope, and a sudden break or pulling out occurs. 

Besides the greater life of the rope, and consequent less 
cost, the saving of time on account of fewer breakdowns 
and stoppages is a factor worth considering; and although 
this feature does not cut as much of a figure with rope- 
driving as it does with factory belting, yet it is of such 
importance that many men would rather use some other 
form of transmission than suffer the annoyance incident to 
respl icing a broken rope every few months. While the 
time lost in repairing the rope or in laying down a new one 
may not be great in itself, the stoppage of a department in 
a busy season may prove to be a serious loss. Where 
foundries and isolated shops have been driven by small 
ropes this has happened so frequently that the ropes have 
been taken out and replaced with shafting, or other more 
expensive method of transmission. 

In order to avoid any serious loss of time or inconven- 
ience due to a sudden rupture two independent ropes 
should be used, each having its own tension sheave and 
weight. This does not involve any more wraps than would 
otherwise be used for a single wind, for the normal work- 
ing stress is in any case so much less than the actual 

rength that for a temporary run of a few hours, or even 

'JB, one rope could readily carry double its working load 


in case the other should give out. When two independent 
ropes are thus used they may be wound separately, each 
wrap occupying successive grooves on the pulleys ; or, 
which is more frequently the case, the ropes may be wound 
in parallel, thus bringing each rope in alternate grooves: 
in either case it is preferable to use an independent 
tightener, although a single-tension carriage, provided 
with two sheaves, may be used; but with this latter, owing 
to local causes and the difficulty of splicing both ropes of 
an equal length, the load is not as well distributed between 
the two ropes. 

In subsequent considerations of the driving power of 
ropes the relation between the ultimate strength, weight 
per foot of length, normal working strain, and the diame- 
ter of rope will be represented by the following equations 
which have been determined for manilla transmission rope : 

Let d = diameter of rope in inches; 

w = weight of rope in pounds per foot; 
8^ = breaking strain in pounds; 
t = normal working strain in pounds; 
a; = an empirical coefficient. 
Then t^ = 0.3216(^'; 
8, = lOOd'x; 
t* = 200^; 
a; = 81 - 9A 

The weight w per foot of length varies considerably in 
different makes of rope, depending upon the amount of 
twist and the foreign matters in the rope. 

It is well known that much of the cheaper manilla rope 

— — «. — . ^ 

* On account of tlie relatively great difference between Si and t, it 
is not thought advisable to consider the increase in strength of the 
smaller ropes, as in any case the difference would be very slight, and, 
moreover, it must be noted that t is the normal estimated strain, and 
may vary considerably from the actual strain. 



on the market is largely adulterated with weighing mate> 
rial, snch as gelatine size, French clay, and white lead. 
Thus some manilla ropes will weigh not more than Oj26(r 
ponnds when dry and very loosely twisteJ; in other cases 
the weight will he as mnch as OAQcP ponnds per foot of 
length. The value we have given, viz., 0.32rf* ponnds, cor- 
responds very closely to the average weight of good quality 
lubricated transmission ropes. 

Cotton ropes are abont twenty per cent lighter for equal 
diameters, and will vary from 0.20d* to 0.29d* pound per 
foot. In the following table (VII) 0.32d* has been used 
for manilla and 0.26^ for cotton ropes. 

Table VII. — Weight of Ropes. 




Weight in Pounds per Foot. 

w = 0:62cP. 

to = 0.96fi». 












Ik determining the horse-power which a rope will trans- 
mit under given conditions the centrifugal force due to tlie 
velocity and weight of rope is an important factor, and its 
influence should be considered in all cases where the 
speed is greater than 2000 feet per minute; for at high ve- 
locities this force diminishes the pressure exerted between 
the rope and the circumference of the pulley, thus reducing 
the friction between rope and pulley. When, therefore, a 
rope has to transmit a given force, P, it must be subjected 
to a greater tension the greater the centrifugal force F^. 
At a speed of about 90 feet per second the centrifugal 
force increases faster than the power from increased veloc- 
ity of the rope, and at 140 feet per second this force equals 
the assumed allowable back tension in the rope; and since 
the transmitting force is equal to the difference in tension 
in the two parts of the rope, it will be seen that no power 
will be transmitted at this speed unless the assumed allow- 
able tension be exceeded. 

It is evident that for a given total tension the less back 
tension required to prevent the slip of the rope on the 
pulley the greater will be the power transmitted at a given 

The determination of this back tension is, however, at- 
tended with a degree of uncertainty, as there are no con- 
clusive experiments which give reliable data for its calcu- 
lation; the coefficient of friction, 0, as stated by various 
authorities, varying all the way from 0.075 up to 0.88.* 

* The probable reason for such widely divergent values lies in the 
fact that the coefficient of friction varies with the percentage of 
slip, and those tests made with very little slip would show a small 



Eeuleaux quotes the experiments of Leloutre and others 
as indicating a value of 0.075 for cylindrical pulleys with 
new hemp rope, 0.088 for semicircular grooves, and 0.15 
for a wedge groove of 60°. 

Experiments by the Messrs. Pearce Brothers, of Dundee, 
give a value of equal to 0. 57 to 0.88 for ropes on ungreased 
pulleys, and — 0.38 to 0.41 when the pulleys are greased. 

Unwin states that the coefficient of friction for ropes on 
a flat-metal pulley is equal to 0.28, from which the actual 
coefficient for a grooved pulley is obtained by multiplying 
0.28 by the cosec. of half the angle of the groove. For an 
angle of 45° this would give = 0.72. These latter values 
are probably very much higher than is ordinarily found in 
actual practice with well-lubricated ropes and moderate 
slip. From a consideration of the above and various other 
experiments, and the conditions under which they were 
carried out, it would appear that for ropes which are partly 
worn and sufficiently greased to wear well with a low per- 
centage of slip a value of 0.12 for a flat-surfaced, smooth- 
metal pulley will approach very closely to those conditions 
which obtain in average practice, from which the following 
working coefficients are deduced : 

♦ =0.18eosec('^"K'«°^«"°™) 


Angrle of ffroove 

30° 35° 40° 45° 
0.46 .40 .35 .31 

50° 55° 
.28 .26 

Coefficient of friction, 0. . . 


Besides varying with the angle of groove, as shown, the 

coefficient; on the other Iiand, it is safe to assume that the larger 
values were obtained under conditions in which the slip was greatly 
increased. Varying atmospheric conditions and different degrees of 
lubrication are also largely responsible for these divergent results. 

See paper by Prof. Lanza on "Friction of Leather Belting," in 
Trans. A. S. M.E., vol. vii. p. 347; also "Experiments on Power 
" »\usmitted by Belting," by Wilfred Lewis, in Trans. A. S- M. E. 
VII. p. 549. 


coefficient of friction is affected by the condition of the 
rope, and for dry ropes (f) may be taken somewhat greater 
than the above value; will also be increased with an in- 
creased percentage of slip. 

If the arc of contact on smaller pulley, the coefficient 
of friction between rope and sheave, and the total tension 
in the rope be known, the tension on slack side of the 
pulley, and hence the horse-power transmitted, can be 
readily determined from the following considerations:* 

Assuming, as before, that the driving force P is equal 
to the difference in tension 2\ on the driving side of the 
rope and T^ on the driven side, and noting that the driv- 
ing force must equal the friction F between the surfaces, 

we obtain 

T,- T^ = P =^F. 

The friction F depends upon the arc of contact a between 
the rope and its sheave, the coefficient of friction 0, and 
upon the centrifugal force F^ set up in the rope, due to 
its velocity and weight; it is, however, independent of the 
diameter of pulley. To determine the values of F, T„ 
and T^ it will be necessary to assume a given tension in 
the rope; also its speed and weight, coefficient of friction, 
and arc of contact. 

Let/ = friction between element of rope and pulley; 
g = acceleration due to gravity = 32.16; 
p = normal pressure exerted by element on pulley; 
t = allowable working tension = 200^^ pounds; 
V = velocity of rope in feet per second ; 
w = weight of rope per unit length and area 
= 0.3216^?" pounds per foot; 

z = abbreviation for — 7 ; 

9 t 

* Weisbach, vol. in. p. 254. See also Reuleaox. 


A = area of cross-section of rope; 

JP^ = centrifugal force due to speed and weight of 

P = driving force = T, — T,; 
A* = radius of pulley; 
T = tension in rope at any point; 
T, = tension in rope on tight side; 
T, = tension in rope on slack side; 
Of = least arc of contact between rope and pulley — 

circular measure = 0.0175 X arc in degrees; 
= coefficient of friction. 

If in Fig. 4ii T is the tension in the rope at any point 
2), then the tension at the point E, whose distance from 
D is (fo, wiU be T + dT. Assuming / to represent the 


Fig. 43. 

friction on the element DE^ we shall haTe/= dT when 

the several forces are in equilibrium. Since the friction 

is dependent npon the normal pressure/^ exerted by the 

element upon the rim of the pulley, and since this normal 

vare is diminished bv the ceiitrifusfal force due to the 

htMzid yelocitv of the element, we shall hare 


dT=<P{p-F,) (4) 

Now jt? is the resultant of the two forces T'and T+ dT; 

which may be assumed equal to Tdoc on account of the 
smallness of da and dT, 

The centrifugal force of the element of the rope is 

F, = wA^ds = xoj^ds; 

and since ds = Rda, we have 

T v^ 
F^ = 70— —da. 

^ SI 

Substituting these values of jo and F^ in dT= (f>{p — F^), 
we obtain 

dT = ^ (Tda -- w^- jda^ ... (5) 


For convenience, let z = — -; then 


dT = (pT{l - z)da, or ^ = 0(1 - z)da. 


'^' dT 

= / 0(1 — z)da, 


^. ^ g«.(l - z) ^j. y ^ y 

hyp log -^' = 0(1 - z)a; ... (6) 

• 9 



in which 6 is the base of the hyp log = 2.7183; therefore 


y _ 2; = p = 7;[€*^^-^>] - ^, = 


If we assume -^ = r, there is obtained 

r y. T. 

r-l~ T,-T,~ P' 


These ratios, r and -, Eeuleaux calls the friction 

* r — 1 

modulus and the stress modulus, respectively. 

As the common logarithm of e is 0.43430, the value of 
r may be more readily obtained from common log r = 
0.4343 0«(1 — z)\ if the are of contact is given in degrees, 
a = 0.01T5a°, which gives the common log 

r = 0.4343 X .0175a 0(1 - 2:) = O.OO:5:S0a (1 - z\ (8) 

As the weight of a manilla rope one foot long = 0.32^* 
pounds, the value of z for varying speeds can be deter- 
mined from 

icv" 0.01<rer» 

'=7^=—/— <^> 

If now we assnme a constant working stress / = SOOiT 
pounds, then 


' = ^^^r/-"- = 0AHXK>5r' (10) 

In the work which follows wo siiall asi>ume that the ten- 
sion 7*. in the rope on the tiiiht side (driving tension) ev^uals 
th.^ itlowable tension /• 

From these assuuiptioJis the folio w:r:g table (VIII) of 
the values of 1 — -s has Iven oomi>uteil: 

BOPE-DKTVlKa, 117 

Table VIII.— Values op 1 — g for a Wouking Stress equiva- 
lent TO 200d* Pounds. 

per Minute. * ~ ** per Minute. ^ ** 

1000 0.98 5500 0.58 

2000 0.94 6000 0.50 

2500 0.91 6500 0.41 

8000 0.87 7000 0.32 

3500 0.83 7500 0.22 

4000 0.78 8000 0.11 

4500 0.72 8500 0.0 

5000 0.65 

It will be seen from the above that when the velocity of 
the rope is as great as 8500 feet per minute, 1 — z = 0, 


hence log r = and -=7 = 1; that is, T^— T^ = 0, and 

therefore no power will be transmitted unless the assumed 
working tension t be exceeded. 

In average work the lesser arc of contact embraced by 
the rope — generally on the smaller pulley — will be about 
165°, and this value may be assumed for appioximate cal- 
culations with a working degree of accuracy. If the angle 
in degrees is known, its value, a, in circular measure, can 
be obtained from Table IX, in which a = 0.01 75a°. 

Table IX. — Angle EoiSRACED bt Rope. 

Fraction of 


Circular Measure, 
































If we now assume the coefficient of friction to be 0.31 
for a 45° groove, we may obtain the value of the ex- 


^=€*^-'>=:r and 5- 

fp - — p r-r 


In order to simplify calculation, the following table (X) 

contains yalaes of r and -^ which wiU enable the horse- 

r — 1 

|K)wer transmitted by a rope to be determined with a degree 
of accnracy depending upon the assumption of the coeffi- 
cient of friction: 

Table X. — ^Fkiction Aim Stress Moduli. 

^l-z). r = ^. 731 = 7^ 

0.1 1.11 10.41 

O.a 1.23 5.40 

0.3 1.35 3.80 

0.4 1.49 3.03 

0.5 1.65 2.54 

0.6 1.83 2.33 

0.7 3.01 1.99 

0.8 ^ 3.33 1.83 

0.9 3.46 1.69 

1.0 3.73 1.58 

1.1 3.00 1.50 
1.3 3. S3 1.43 

1.3 3.67 1.37 

1.4 4.06 1.33 

1.5 4.48 1.39 

The following appliaition will show the use of the tables. 
Let it be required to determine the horse-power transmitted 
by a rope 1 inch in diameter running at a velocity of 4000 
feet per minute orer a pulley with 4o^ grooves. Assuming 
an arc of contact of li>5% we find from Table IX a = 2.S8; 
for the required Tekvity, 4000 fet^t [vr minute. Table YIII 
gives 0.7S as the value of 1 — i; therefore, assuming the 
coefficient of friction (p = .31.. we obtain 

0a(l - c) = 0.31 X :J.SS X -TS = .69. 
From Table X the value of -^. corresponding to 0,69, 



is about 1.99; and as T^ = 200d^ pounds = 200 in the 

present case, we find P = ^-^ = 100 pounds. Since 



100 X 4000 

33000 = ^' P- *^''^ '^ ^^^^"'^ ^- P- = 33000 ^ 
12,15, represented by the ordinate Im in Fig. 43. 
The loss due to centrifugal force may now be obtained 



9 S 








O 10 

00 CO 

Fig. 43. — Centrifugal Effect in Ropes. 

by assuming the latter reduced to zero, in which case the 
factor 1 — 25 is equal to unity; therefore log r = .4340a, 
from which, with the previous conditions, we obtain P = 
120 and the corresponding h. p. = 14.5. 

This value is represented on the diagram. Fig. 43, by the 
ordinate hi : the difference between In and Im — mn will 



then be the loss dne to the centrifugal force set up in the 
rope = 14.5 — 12.15 = 2.35 horse-power. 
For any special case where the data are known or may 



O O 10 

h<. 00 00 

Fig. 44. — ^Horse-power transmitted by Manilla Ropes. 

be determined, the formulas and tables already given should 
be used to ascertain the horse-power transmitted, or the 
diameter and number of ropes required for a certain work, 
as the case may be. For average work, however, it will be 



found that the assumed values of a and 0, previously 
noted, will give very siitisfactory results, and upon these 
assumptions the writer has computed the following table of 
horse-power (Table XI) for various-sized ropes, running at 
speeds from 1000 to 7500 feet per minute: 

Table XI. — Horse-power transmitted by Ropes. 

Working Strain = 200d' pounds. 

d = diameter of rope in inches. 

Velocity of 

Rope in Feet 

per Minute. 

Diameter of Rope. 
















































































































The graphic representation of these values. Pig. 44, 
shows the effect of centrifugal force in diminishing the 
power transmitted under an assumed working tension, and 
would indicate that with tensions of 200^?" pounds the 
speed should not exceed 5500 feet per minute. The in- 
creasing effect and loss of power due to centrifugal force 
in the rope can also be seen in the diagram, Fig. 43, which 
represents the horse-power transmitted by an inch rope 
under an assumed constant tension of 200 pounds. The 
straight line AB shows tlie power which would be trans- 
mitted if centrifugal force were neglected, and is obtained 
by making jk = in the general equation 


log r = 0.4343^0(1 — z); 

the curve AC represents the power transmitted when cen- 
trifugal force is taken into account; and the curve AB 
shows the power absorbed by centrifugal force: this latter 
curve is obtained by subtracting the vertical ordinates be- 
tween the straight line AB and the curve AC, By a con- 
sidation of the diagram it will be seen that at speeds 
less than 2000 feet per minute the power absorbed by 
centrifugal force is very small, and may be neglected as 
far as practical results are concerned. Beyond this speed, 
however, the loss from this cause increases very rapidly, 
until, as previously shown, at a speed of about 8500 feet 
per minute the whole of the allowable tension is absorbed. 

Assuming that the maximum power is transmitted by a 
rope at a velocity of about 5500 feet per minute, it is evi- 
dent that the first cost of the rope will be a minimum for a 
given power when running at this speed. The ratio of the 
first cost of the rope running at any other speed may be 
obtained by dividing the horse-power at 5500 per minute 
by the horse-power at the required speed.* 

Thus, if the first cost of a IJ-inch rope which will trans- 
mit 21.14 h. p. at 5500 feet per minute be represented by 
unity, the cost at 3000 feet per minute will be 

15:39 " ^'^^' 

since a i-inch rope running at 3000 feet per minute will 
transmit 15.39 h. p. 

The relative first cost for a given diameter of rope to 
transmit the same horse-power at varying speeds is shown 
in the accompanying Table XII. 

Although the first cost of a rope to transmit a given horse- 
power is a minimum for a speed of about 5500 feet per 

* C. W Hunt, in Trans. A. S. M. E., vol. xn. 



minute, yet the economy is not as great as would appear from 
the foregoing table, for the effect of wear must be considered. 
The causes of wear, internal and external, have been pre- 
viously discussed; it will be sufficient to note here that the 

Table XII. — Relative First Cost of Rope-driving. 

Velocity of Rope 

Relative First 

Velocity of Rope 

Relative First 

in Feet 

Cost per 

in Feet 

Cost per 

per Minute. 


per Minute. 


























internal destructive effect produced by bending and distort- 
ing the fibres and the wear due to external contact, slipping, 
or wedging in the grooves of the pulleys, may, within the 
limits of ordinary practice, be cousidered as directly propor- 
tional to the velocity of the rope. What this wear is in 
terms of the velocity there is not sufficient data to deter- 
mine, but if the coefficient be represented by c, the relative 
wear for a given diameter may be determined by multiply- 
ing the velocity by this coefficient; that is, the relative 
wear = cv, in which the wear increases directly with the 
velocity, but not, however, directly with the horse-power 

If we assume the coefficient to be such that the wear on 
a rope at 1000 feet per minute is equal to unity, then the 
wear on the rope at any other speed will be 

^ , .. required speed 
Relative wear = — - — ^ . 

. To determine the relative wear per horse-power trans- 
mitted by a given rope at varying speeds, it will be neces- 


sary to determine the wear per horse- power for the rope 
running at any required speed, and then divide this value 
by the wear per horse-power when running at 1000 feet per 
minute. Let it be required to determine the wear of a rope 
transmitting a given horse-power at 5500 feet per minute, 
as compared to what it would be at 1000 feet i)er minute. 

The horse-power transmitted at 1000 feet per minute is 
found to be 3.57 for a one-inch rope, at which speed we 
have assumed that the wear is equal to unity, hence the 
wear per horse-power may be considered equal to 

= .28. 


At a speed of 5500 feet per minute the horse-power trans- 
mitted by the same rope is 13.53, but the wear at this 
speed is assumed to be five and a half times greater than 
at 1000 feet, other conditions being the same, therefore 

the wear per horse-power = ,-oTq = .406; that is, the 

wear of any rope transmitting one horse-power at 5500 feet 

per minute is ^-^r^ = 1,45 times the wear which would 

occur at 1000 feet per minute. 

From this it will be seen that although the first cost of 
a rope is cheaper at the higher speeds, the rope lasts longer 
while running at the lower speeds — conditions remaining 

Taking the case we have been considering, it is found 
that the relative first cost of the rope is inversely as the 
horse-power transmitted, or 

Cost of rope at 1000 feet __ 13.53 _ « ryo. 
Cost of rope at 5500 feet "" "07 "" ' 

that is, the first cost is 3.78 times greater for the slower 
tpeod. But it is shown above that the rope will wear out 


J 25 

nearly 50 per cent faster at the increased speed; therefore, 
taking the life of the rope as well as the first cost into con- 
sideration, the relative cost to transmit a given horse-power 

3 78 
at the speeds noted will he -^-— = 2.6 times greater for the 

lower speed. This can be determined more conveniently 
by assuming ^ = horse-power transmitted at F feet per 
minute, H^ = horse-power transmitted at F, feet per min- 
ute, where H^> II and F, > F. The wear per horse- 

F F 

power in each case will then be proportional to -yj^and -^; 


the relative first cost of the rope per horse-power running 
at the lesser speed, compared to that when running at the 
greater, will be as H^ is to S; hence the ratio of relative 
cost, taking wear into account, is 


H V^ V\h) 


Table XIII.— Relative Wear and Cost of Rope per Horse- 

POWER transmitted. 


in Feet 

per Minute. 

Relative Wear 

per Horse-power 


Relative Cost of Rope 

per H. P. transmitted, 

considering Wear. 



















Table XIII has been calculated upon the above basis of 
comparison, namely, that the wear in a rope at 1000 feet 
per minute is equal to unity arid increases directly as the 
speed; and also, that the cost of ropes for a permanent 






o o o o 

2 8 S o 

o 5 o o 

CO "t lO (D 

\f< '»-irv <N rcrr rr- ••INUTr. 


rs Q) 

Fig. 45. — Relative Wear and Cost op Rope per H. P. 

installation is proportional to the square of the ratio of the 
power transmitted at different speeds multiplied by the in- 
verse ratio of the corresponding speeds. To ascertain the 
relative wear per horse-power of a rope running at any 
given speeds, it will only be necessary to form a ratio 


between the values in the table corresponding to the given 

speeds. Thus the wear per horse-power at 5000 feet per 

minute, as compared to that at 2500 feet, will be -^ = 1,26. 


In the same way the relative ultimate cost per horse- 

power of a rope running at these speeds will be '-jz = 0,8, 


or 20 per cent less for the greater speed. The accompany- 
ing diagram. Pig. 45, represents these relative values 
graphically, to which is added the curve of relative first 
cost. It will be noticed that although the first cost of a 
rope is a minimum for a speed of 5500 feet per minute, 
when wear is considered the minimum cost occurs at a 
speed of 4500 feet per minute. 



As previouj;'/y noted, it is desirable in all cases of rope 
transmission to ;?o arrange the drive that the slack side of 
the rope shall be on the upper part of the pulley, thus in- 
creasing the a^*c of contact, as the two sides will then ap- 
proach each other when in motion. 

In order that the desired tensions 7\ and T^ shall be 
attained in the two parts of a rope, the deflections or sag 
must be of predetermined values. The centre line of the 
rope will lie in a curve, which may be determined with no 
appreciable error by assuming the rope to have no elastic- 
ity and to be of constant cross-section, under which condi- 
tion the curve will be that known as the catenary, the trans- 
cendental equation of which is 

^ / as x\ 

y = ^[^ + e'^} (12) 

Let the form of curve in which the rope hangs be repre- 
sented by PO'P' (Fig. 46), in which 0' is the lowest point 
of the rope. Take 0' as the origin of coordinates, and let 
X = 0'-4' and y' = A'F be the abscissa and ordinate of 
any point F in the curve, and let I be the distance between 
the points of support. 

Since we have assumed the rope to be perfectly inelastic, 
the tension at any point of the curve must be in the direc- 
tion of the rope. Let T be the stress in the rope at P, the 
vertical and horizontal components of which are repre- 
sented by V and H respectively. Assume the length of 
curve O'P = s. Since the weight of a unit length of rope 
= to, the vertical component of the tension T is evidently 



equal to one half the weight of the rope between the points 
P and P', or V—^w PO'P' = ws. To determine the 
length of curve, produce the tangent through the point P 
and rectify the curve O'P on this tangent, making s = PQ. 


Fig. 46. 

Erect a perpendicular to PQ at the point Qy meeting the 
vertical from P 2X A; then will OA drawn through A par- 
allel to O'A' be the directrix to the catenary, and QA will 
equal AA' ^ c* Hence 

Pq^ = PA^ - qA\ 
Therefore V ■=^ws = w V {y'Y + 2cy\ 

* For geometry of this curve see ' * The Funicular Polygon " in 
Bowser's ''Analytic Mechanics,*' p. 216 et seq,; also Price's "Me- 
chanics," vol. I. 


At the vertex the tension is liorizontal and eqnal to the 
weight of a lengthy c, of the rope; but this tension is the 
same as the horizontal component at the point P; hence 

H =^ wc. Since 7'= VV -\- H*, we obtain the following 
value for the tension in the roper 

T=wVy'^-^^cy' + c^-w(c + y'). . . (13) 
In order to determine the parameter, c, let the equation 

6f the curve ^ =- ^e^ + e 7 be developed into the following 


series : 

c (^ , X . «• , x* X 

y = 2l^+^+r:27'+r:2T3^ • • • +^"^ 

X^ _ X* \ 

■*"l7y? 1.2.30* ■*" /• 

Since the character of the curve is such that the quotient 


— is a proper fraction, the series will be converging. Stop- 

ping at the third member as giving sufficient accuracy, we 


Iiherefore a? ^2c{y — c) (14) 

Substituting the value oiy = y' -^cin this equation, we 
obtain m? = 2cy', which is the equation of a parabola re- 
ferred to its axis and the tangent to its vertex. Now let ~ 

(Fig. 47) equal the half distance between supports = a:, 
and A = sag of the rope = y' ; then from the previous 

equation we obtain ( o* ) = 2cA: 

hence the parameter c = — . Substituting this value of 



c in equation (13)^ we have 


in which w equals the weight of a unit length of rope = 
0.32^^ for manilla ; that is, the tension at any point in the 
rope is equal to the weight of a portion equivalent in length 
to the parameter plus the ordinate y' of the point. From 
this equation we may obtain the sag of the driving or driven 
portions of the rope by substituting for T the values of T^ 
and 7\,the deflections corresponding to which may be rep- 
resented by A, and h^ (Fig. 47). T^ will evidently be con- 

bmvEN "~ 


Fig. 47. 
stant and equal to 200^/^ if our preconceived conditions 
are maintained, but the value of 7\ will be variable, in- 
creasing with the speed. 

Practically, it will be impossible to maintain a constant 
tension in the rope, so that the amount of sag obtained by 
calculation is liable to varv with the conditions of service. 
The tension may, however, be approximately determined 
by the deflection. Assuming the distance between support- 
ing points of the rope equal to the distance between centres 
of pulleys, and solving for A, we obtain 

*' = 2^-=^'2r ^-25 

IT 1 /~T^ 7« 


The positive and negative signs before the radical in these 

equations indicate two values for /*, the lesser of which 

only is to be used. As pointed out by Reuleaux, between 

the two lies a value h = - -, which is obtained when the 


quantity under the radical = 0; that is, when r=-— • 

Tliis deflection is interesting, as it denotes the minimum 
stress which may exist in the rope. 

Since the sum of the tensions increases with the speed, 
the sag of the rope when at rest is not directly obtainable 
from the f evious values of h^ and 7^,. 

The tension ii3cessary for adhesion, which constitutes a 
part of the stress, jT,, is dependent npon the speed at 
which the rope is intended to be run, so that in order to 
determine the sag //„ when at rest, for a given maximum 
tension T^, the initial tension must be obtained for any 
given speed, and this value substituted in the general for- 

2tv^ ?y to* 2' 

Assuming the tension on the tight side of the rope to 
be made up of three parts, namely, the driving force F, 
the centrifugal force F^, and the tension T^, necessary to 
balance the strain for adhesion, we obtain 

T,= P + F,+ T,. 

In like manner the tension on the slack side of the rope 
may be assumed to be produced by the strain necessary for 
adhesion, plus the strain due to centrifugal force; that is, 

T,= T,+ F,. 

It is evident that if the normal tension 7\ be diminished 
by the centrifugal force the remaining stress will be equal 
to the tension necessary for adhesion plus the driving 




force; hence if we obtain the value of the ratio ~ from 

€^*, in which the stress due to centrifugal force is neg- 
lected, we shall have in T^ the initial tension necessary for 
adhesion, or T^= 7\. If 2\ is assumed to be constant, then 
T^ will be constant for a given coefficient of friction* and 

arc of contact; therefore 


will equal the tension in 

the rope when the latter is at rest. 

The following simpler method for obtaining the sag, 
though less exact, is sufficiently accurate for any practical 
case that may arise, for it must be borne in mind that any 
theoretical calculation for the deflection of » running rope 
can at best be only an approximation, as it is exact only 
when the rope is running at its normal speed, transmitting 
its full load and strained to its normal tension. Let I be 
the distance between two shafts which are at the same 


CD = 2h 
Sin 0=CD 




Fio. 48. 

level (Fig. 48), and let h be the deflection of the rope; also, 

let CPD = 6^ be the inclination of the rope at P, from 


2h 4A 


and hence, since Fis the vertical component of the tension 




T in the rope and equal to — , where W is the weight of 
rope between supports, we shall have 


sm o Sh ^ 

64A* + 4^'' 

As h is small compared to Z, we may neglect — ^*, as it 

will have no appreciable influence on the result; then if 
we assume the length of rope to be equal to the distance 
between the shafts, we shall have, approximately, 

8/i ■" 8/i' 

where wl = JT; hence the sag of the rope at the centre 
will be 

A = ^, [15] 

in which w = 0.32^' for manilla ropes. 

From this formula the following table (XIV) has been 

Table XIV.- 

—Deflection of 


Calculated from 7a — ^^ 

Deflection in Feet in Slack Side of Rope 

in Both 

when Ti = Ti-P = 








Tight Side 

of Rope 


3\ = 200da 

Sides of 
the Rope 

in Feet. 


sponding to 
per Mir 


Velocity in 
lute of: 


I Feet 






























































computed ftnd the curves plotted, as sliown in Fig. 49, 7", 

being assumed its constant and oqiial to 200rf pounds. 

Tlie tension 1\, which varies with the speed, has heen 

separately determined fov the several cases considered from 


ti — e*^i-*)^ 

in which the coeflBcient of friction, 0, =0,31 and a = 2.SS: 


40 aO 80 100 120 140 160 Ft 

1. 49.— Dkpleci'ion op KoPKa. 


log ^ = 0.4343 X 0.9(1 - z). 
The deflection on the tight side will then be 

h = 

Mr=8^^20(r^^ = ^-^^^^^' 

and on the slack side A,= ' ^ , in which T^ has the values 

given in the table. 

As previously pointed out, the initial tension in the rope 

when at rest may be obtained by neglecting F^ , in which 

case we have log ^* = O.43430ar, from which we find 

'^ = 2.46, or T; = 82flP; and since T, = T, when there is 

no centrifugal force acting, the initial tension T^ will 

y,+ t; _ 200^' + 82^^ ^ 

2 " 2 * 

hence T^ = 141^?'. From this we may obtain the deflec- 
tion when the rope is at rest, as noted in the last column 
of Table XIV, which has been computed from 

K = ^^^ = 0.000284P, 

where I is the distance in feet between centres. 

To draw the curve of the rope in order to determine 
the space it will occupy, assume it to hang in a parabola 
with origin at vertex and lay off the X and l^axes, as in 
Fig. 50. Let 05 = il = i distance between points of sus- 
pension of the rope, and let 5^ = h, the sag of rope. Di- 
vide 5H into a convenient number of equal parts, also 
divide 05 into the same number of equal parts; erect per- 
pendiculars from the points of division 1, 2, 3 ... and join 



1', 2', 3'. . . with the origin 0, The curve drawn through 
the points of intersection will represent one branch of the 
parabola desired. 

In outdoor drives, where the configuration of the ground 
would prevent the proper amount of sag being used, or, in 

4 5 . 


Fig. 50. — Method of Laying Out Curve. 

• flee. 


Fig. 51. — Inclined Transmission. 

general, where obstructions prohibit the proper deflection 
for a given distance between centres of shafts, the rope 
should be carried on supporting pulleys or sag-carriers, 
examples of which have already been given. 


Where the pullejs are placed at ilifFerent heights we 
have an inclined amission, iiml the curve In such cases 
is unsjm metrical, as in Fig. 51,* 

This curve is best solved bj an approximation similar to 
that already given. 

It will be noticed that the curve of the rope ABC is 
made up of two unequal piirts, AB and BC, whose hori- 
zoutal projections are /' and I" and their corresponding 
doflections A' and A". The given difference in height be- 
tween the iMtints of support of the rope is SD = A ^ h' — 
/(", and their known horizontal distance is I = I' -\- i". 
From the preceding formula for the deflection (equation 
[15] ) we have 

* sr ^ ^' *""* * 87^ = ^' ' 
in whicli fi is taken aa a coefhcient depending upon the 
value of the tension in the rope = -^yp- 
Since A = A' — A" aud ? = /'-[- I", we hsive 
h = nr - V) = fllf - l")(V + 1") = 0l(V - !"); 

therefore V — I" = -z,-,. But V = 1 — I" and I" = 1 — I'; 
hence 3/SW — /Si' = A, and the required distance V = 

'^ ■ Ti. (Via comawaTT 7" — ^ ^jj^J Jde 

4' = <!r-^^?^ and 

ill u:ich of wlii<.-h equations ^ has a separate value depend- 
ing upon the tension in the rope. It is evident that the 
tension T' iit A will be greater tlian that at G, on account 
of the greater weight of rope lying in the upper briinch ot 
• WeUbach, vol. al. p. 343. 



the parabola. If T' is known, the lesser tension T" can 
be determined by assuming the length of the rope to be 
the same as if the points A and C, Fig. 52, were at the 
same height and I feet apart. From previous considera- 


Fig. 52. 

tion it is seen that tlie tension at any part of the rope in 
a parabola is equal to the weight of rope equivalent in 
length to the parameter c added to the ordinate of the 
point; hence if h" equals the ordinate of C = h' — /i, we 
shall have 

r ' = 'w{c + A") and T' = zo{c + V). 
Since c = L 

therefore T" - wA" = T' - wh\ Substituting for 7/ its 
value, ?t^ + ^^"> there is obtained 

7 ' = 7 



When a tension-carriage is used the necessary weight 
can be ascertained from the formula for back tension, 


T^==T^-\-F^; if we assume F^ = 0, we shall have the 
initial tension necessary for adhesion equal to the ten- 


sion in slack side of rope, that is, T^ = T^. But T^ = — ^ 


(from log -^ = 0.43400', assuming previous values of 

and a); therefore the initial tension in the rope will be 

'""2.46"" 2.46 • 
When diameter of rope equals 

f inch the initial tension = 31 pounds. 






= 45 







- 62 







= 82 







= 127 







= 184 














- 325 


The actual weight to be placed on the tension-pulley will 
depend upon the arrangement of drive, and in order to 
obtain the best results should be determined for each par- 
ticular case. It is to be noted that with a vertical tension- 
carriage the weight of carriage and sheave must be consid- 
ered as a portion of the weight producing 
' \ ^ 2fl ^ / ' tension in the rope. 

If W equals the total weight necessary 
to maintain the tension T^ in each part 
of the rope leading off from the tension- 
pulley, then, when these two portions are 
parallel, we shall have 1^=2 7^3; but if 
the rope leads from the tension-pulley so 
that it includes an angle 2ff between its 
sides, as shown in Fig. 53, then the weight W will be less, 
and may be found from If = 2 7^, cos 0. 



It has been previously stated (see page 7) that where 
rope-driving is used the loss at the engine in friction may 
be taken in a general way as about 10 per cent of the rateci 
horse-power of the engine, that an additional 10 per cent 
is absorbed by the shafting, and that from 5 to 8 per cent 
may be attributed to losses in the rope itself, due to resist- 
ance to bending, wedging in the grooves, differential driv- 
ing effect, and creep, all of which affect the loss to a greater 
or less extent. 

According to the accepted laws of solid friction we should 
expect that with an increased load on the engine the fric- 
tion would be increased in direct proportion to the load, 
but in nearly all experiments to determine the friction of 
steam-engines we have the anomaly that the work neces- 
sary to overcome friction is practically constant, and, in 
fact, in many cases it is a little greater when running with- 
out load than when the engine is fully loaded. 

However, it is probable that the ordinary laws of friction 
obtain here as in other cases of sliding and rolling con- 
tact; but instead of having a constant coefficient of friction 
for the surfaces in contact, the coefficient may be con- 
sidered as a variable depending upon the degree and distri- 
bution of lubrication. The lubrication of engine bearings 
(both rolling and sliding contact), approaches more nearly 
to the condition which obtains when the bearings are sub- 
jected to an oil bath, and although the lubrication is 
restricted, yet the result is similar in action. 

Experiments* on the friction of a well-lubricated journal 

*Proc. I. M. E. November, 1885; also Kent's Mechaniciil Eugineer's 
Pocket Book, 


(oil bath), show that the absolute friction, that is, the ab- 
solute tangential force per square inch of bearing, required 
to resist the tendency of the brass to go round with the 
journal, is nearly a constant under all loads within ordinary 
working limits. Most certainly it does not increase in 
direct proportion to the load, as it should do according to 
the Ordinary theory of solid friction. The results of these 
experiments seem to show that the friction of a perfectly 
lubricated journal follows the laws of liquid friction much 
more closely than those of solid friction. They show that 
under these circumstances the friction is nearly indepen- 
dent of the pressure per square inch, and that it increases 
with the velocity, though at a rate not nearly so rapid as 
the square of the velocity. 

The experiments on friction at different temperatures 
indicate a great diminution in the friction as the tempera- 
ture rises. Thus in the case of lard-oil, taking a speed of 
450 revolutions per minute, the coefficient of friction at a 
temperature of 120° is only one-third of what it was at a 
temperature of 60°. In regard to engine friction, whatever 
be the cause, it is a well-known fact that the coefficient of 
friction decreases as the load increases, so that at all ordi- 
nary speeds the internal resistance of the engine maybe con- 
sidered sensibly constant, in which case the so-called fric- 
tion-card of the engine represents practically the friction 
01 the machine when fully loaded — the indicated power 
without load being sensibly the measure of the wasted work 
of the engine when in operation under load of whatever 

That is, the engine friction is independent of the load 

and is a function of the characteristic of the engine itself, 

of the speed of piston and rotation, and, to a slight extent, 

'^f the steam-pressure and of the method of steam-distri- 

n : so that while we may speak of the friction as being 

ain percentage of the horse-power of an engine, it 



must be understood to refer to the rated indicated horse- 
power ;* at less than rated power the percentage of loss due 
to friction will be greater, and at maximum power the per- 
centage will be less. This is shown by way of illustration 
in the following table (XV), which is taken from a paper 
presented by Prof. Thurston before the American Society 
of Mechanical Engineers in 1886 :f 

This engine, a "Straight Line,'' was 8 inches in diameter 
of cylinder by 14 inches stroke; it had a balanced valve 
with stroke varying from 2 to 4 inches according to position 
of governor and eccentric, a fly-wheel 50 inches in diam- 
eter, weighing 2300 pounds. Its rated load was 35 to 40 

Table XV.— Friction Per Cent under Varying Loads. 











per cent. 







































































Examining the above table, it will be seen that the fric- 
tion of the engine varies somewhat with varying steam- 
pressures and total power, but in such a manner as to indi- 
cate the controlling cause, as, for instance, imperfect lubri- 

* Thurston, Trans. A. 8. M. E., vol. x. p. 110. 
f Trans., vol. vili. p. OC 


cation, to be irregular in action^ aiid^ possibly^ to some 
extent, due to errors of observation and to accident. Tlie 
average friction horse-power is 3.11 h. p., and the variations 
from this value are distributed throughout the whole series, 
showing that the work necessary to overcome the friction 
is practically constant, and independent of the load. The 
friction of this engine is quite low, as at its normal i-ating 
the percentage of loss is less than 7 per cent. According 
to D. K. Clark,* the frictional resistance of steam-engines 
varies from 8 to 20 per cent of their normal indicator h. p. 
— the size of engines experimented upon ranging from 13 
to 350 h. p. 

In recent tests of American engines it has been shown 
that with first-class workmanship and balanced valves the 
percentage of loss at normal working load may be reduced 
to about 6 per cent, both with Corliss and with high-speed 
single-cylinder automatic engines; but tliis is exceptional, 
and we may expect with ordinary lubrication that the fric- 
tional resistance will vary from 8 to 13 per cent of the 
normal load. 

Compound engines of the better class should not absorb 
more than about 10 per cent, and triple-expansion engines 
no more than 12^ per cent, under full load. 

For small engines, either single or compound, there is 
probably little difference between the internal resistance, 
whether geared with ropes or flat leather belts, for the 
weights of fly-wheel and grooved pulley and the diameter 
of shaft would be essentially the same in each case. With 
larger engines, however, the belts would require a wider- 
faced fly-wheel, which, on account of the greater distance 
between bearings, would necessitate a larger shaft, and 
hence increased work in overcoming journal-friction — as- 
suming the same weight of wheels and speed of rotation. 

* " The Steam Eogiue," vol. ii. p. 616. 



With the rope-driving, also, the speed of the rim will be 
greater, and a somewhat lighter pulley may be used to 
insure the same degree of steadiness in running; in many 
cases the ropes are delivered from the fly-wheel in a nearly 
vertical direction, so that a certain portion of the weight 
on the bearings is neutralized by the upward pull of the 
ropes. Moreover, the elasticity and recoil of the ropes act 



Fig. 54. 

in the same manner as mass in the rim, and for this reason 
a lighter wheel may be used with rope-driving provided 
the construction is such as will permit it. The journal- 
friction is therefore presumably less in the larger class 
of engines employiiig ropes, when compared with those 
using belts. The difference is, however, not great; and 
since the actual resistance in any case is also dependent 
upon peculiarities of type and construction, the lower 
values of engine-friction previously given, viz., 8 to 10 per 
cent of normal horse-power, may be considered to hold 



good for rope-driving plants of medium size, while 9 to 12 
per cent of the normal horse -power may be taken as suit- 
able for the larger class of engines running under favor- 
able conditions. 

In the ordinary transmission of power by shafting we 
find the shaft loaded with pulleys and the power taken off 
in varying amounts throughout its entire length; it is un- 
usual except in short lengths to receive the power at one 
end and transmit it at the other. Moreover, in long shaf t- 







Jlee. World 



Fig. 55 

ing the head or receiving shaft is usually situated midway 
between the ends, and the power distributed more or less 
uniformly from this head shaft to either end; therefore, 
in estimating the power absorbed by friction in ordinary 
mill or factory shafting loaded with pulleys the previous 
formulae (page 70) do not apply, as these relate only to 
those cases where power is taken off at the end of the shaft. 
The conditions of practice as we find them in actual 
transmissions are so various, that it is difficult to lay down 
any general rule by which the power absorbed by friction 


tnay be determined: the number and weight of pulleys 
and couplings, the intensity and direction of belt-pull, the 
condition of bearings and their lubrication, all affect the 
amount of work lost in friction. 

For the ordinary factory shafting, from which power is 
taken fairly uniformly throughout its length and dis- 
tributed horizontally to counter- or auxiliary shafts situ- 
ated on one or both sides of the main shaft, there will be 
three general cases to be considered, as shown in Figs. 54, 
55, and 56, and each of these cases will be modified, depend- 
ing upon the direction of the belt to and from the main 

For our present purposes it will be sufficient to take that 
case in which the shaft friction is a maximum for the 
assumed direc^ion of main belt-pull corresponding to the 
arrangement shown in Fig. 54. 

The friction will evidently be proportional to the weight 
of the shaft and the unbalanced belt-pull acting on the 

The weight of pulleys, belts, clutches, and couplings 
carried by the line-shaft will vary from about one and one- 
half to three times the weight of shaft, so that the total 
weight on the bearings will vary from two and one-half to 
four times the weight of shaft; for head and jaciv shafts 
the total weight will probably vary from three to five times 
the weight of shaft. 

In addition to this weight thero is the unbalanced belt- 
pull, which increases the load on the bearings. Although 
the tension on the tight side of the belt should not ordi- 
narily exceed about twice the tension in the slack side 
necessary for adhesion, yet it is probable that belts ai-e fre- 
quently run with a ratio of tension equal to one to three, 
and occasionally one to four; on the other hand, it is a 
very common thing for belts, especially short ones, to be 
laced so taut that the initial tension is greatly in excess of 



that required for adhesion^ iu which case the sum of the 

tensions approaches twice that in the tight side of the helt* 

With ordinary shop-worn belting it will be safe to assnme 

that the tension 7", on the slack side of the belts is one half 


the tension T, on the tight or driving side, that is, 2^, = — '; 

hence, since T^ — T^ = P, the driving force, we have 
y, _ _ 7\ F 

"2 '~ ^' ^°^ ^^ " "2 ^ SSbOO' 
The velocity of intermediate belting is so variable that 














Fig. 56. 

any assumption of speed must be regarded as applying to 
a particular case or representative of a certain type of 
factory, and cannot be taken as general. In many ma- 

*This is often a source of much trouble, as the increased tension 
not only increases the loss due to friction but in many instances the 
us(jful power transmitted is not sufficient to drive the machine. In 
such cases, by slacking out the lacing or inserting a short piece of belt 
so as to reduce the tension, heavy cuts can readily be taken when it is 
practically impossible to run the machine empty with the tight belt. 


chine-shops the average speed of iutermediate belts is not 
more than 500 feet per minute ; in others the average 
speed is more than twice as great, and in wood-working 
shops it is still greater. 

For our present purpose we shall assume an average 
speed of 660 feet per minute for belts running from the 
main shaft to a secondary or counter shaft. 

T V 

Substituting this value in HP = — i x ^ttt^t:^ there is 

/v Of>UuO 

obtained T, = ^ \ll^^^ HP = 100 X HP-y but since the 
' 660 

horse-power which the shaft is capable of transmitting 

may be considered equal to — — , where d is the diameter of 

shaft in inches and JV'the number of revolutions per minute, 

we have the tension on the tight side of all the belts 

2T^ = 100 X Y^ = d^N; therefore the sum of tensions 

^{^i + T^) = 1^'^^ and the pull per foot of length of 




2L ' 

In the present case it will be noted by reference to Fig. 
54 that there is an additional pull on the bearings due to 
the tensions in the belt from fly-wheel to main-line shaft. 
If the ratio of tight to slack side tension remains the same 
as before, and we consider that the velocity of main belt is 
four times as great as that of the intermediate belting, the 
additional belt- pull will equal approximately one fourth of 
the sum of the belt-pulls from the main to the counter 
shafts or machinery. The resultant of these tensions, 
combined with the weight of shafting and pulleys, will be 
the effective load on the bearings. 

Assuming an angle of 30° with the horizontal for the 
line of action of the resultant pull on the bearings due to 


150 KOPfi-DRlVIKG. 

the tensions in the tight and slack sides of the main belt, 
the combined horizontal pull on the bearings will be 

cos 30° X \{ld'^) + 1^^^= l-2(|d-iVr); 

and the vertical pull will be, when W, = weight of loadet* 

W; + sin3r X^-(|rf'iv). 

Therefore the resultant of both horizontal and vertical 
forces acting on the bearings will be 

= x/[w. + sin 30° X -|(|'''^)]'+ [l • Kl'^'^)]'- 

As previously shown, the horse-power necessary to over- 

^ome journal-friction will be HP^ = , where F is the 

force of friction at the circumference of shaft and v is the 
speed in feet per minute of a point on the circumference. 
If the bearing is well worn and fitted to its shaft, the resist- 
ance due to friction will probably lie between the limits 

--0 If and - 01F, where is a coefficient which, from the 

results of experiments on shafting with ordinary lubri- 
cation, we have assumed equal to 0.06, and W is the 
resultant load, in pounds, on the bearings. 
From the lesser of these values there is obtained 

F=:-(PW. (16) 

But we have assumed that the weight of a loaded shaft 
varies from two and one-half to four times the weight of 
shaft; taking an average value of three for line-shafting, 
and noting that the weight of shaft per foot of length 


equals — (3.36(r), we have the friction on a loaded shaft L 
feei long, due to its weight = 

^03 f~ X 3.36^']i. 


Substituting the value of W in formula (16) when the 
belt tensions are taken into account, and noting that 


TFa = 3 - (3.36rf'ii), we have for the total friction load 
^^n^Vl^ X 3.86(PZ + sin 30'' x |(|<iw)]* 4- [l.2(|dW)] 

From the formula for the power absorbed by friction we 

hence the ratio of power absorbed by friction to the hoise- 
power which the shaft is capable of safely transmitting 
will be 

HP = O.Old^N - -^ P"" """^' • • ^^') 

From this expression the following table (XVI) has been 
computed for the given diauieters and lengths of shafting 
running at 100 and 250 revolutions per minute, the belt 
speed for the secondary belts being assumed at an average 
of 660 feet per minute. 

For intermediate belts having a greater average velocity 
than that assumed, viz., 660 feet per minute, the friction 
horse-power for a given number of revolutions will be less 
than that given in the table. Thus if the average velocity 
of cross-belts equals 2640 feet per minute, the horse-power 
transmitted being the same, it follows that the tensions in 
the secondary belts will be one fourth of that obtained with 



the lesser speed; if the main driving-belt have the same 
velocity, the tension in this belt may be considered equal to 
that existing in the intermediate belts: therefore, as the 
velocity of the belt increases for a given speed of rotation 
the sum of the tensions acting on the bearings will decrease, 
and the maximum horse-power transmissible by the shaft 
will be exerted with a decreasing friction loss. 

Table XVI. — Power absorbed by Friction in Line shaft. 

Diameter of 


Percentage of Loss when Length in Feet — 



in Inches. 







' 250 




* . • • 



. • a • 

. . . • 










. • . . 




















14 2 



Line of action of resultant of main belt tensions = 30". 

Velocity of main belt 2640 feet per minute. 

Belts from line shaft are horizontal and run at an average of 660 
feet per minute. 

All the belts are assumed to drive from one side of the shaft toward 
the engine. 

Weight on bearings three times weight of bare shaft. 

If in any case the shafts are belted vertically or at any 
other angle than that assumed, the formula for F will be 
modified accordingly. 

For a head or jack shaft carrying heavier pulleys the 
weight acting on the bearings may be taken equal to four 
times the bare weight of shaft; in which case, other condi- 
tions remaining the same, we obtain 

F= -0^[-^ (3.36(^2)2: + sin 30'' >< ^(2^'^)]'+ [^-^d^^)]'* 

Since, liowever, the extra weight of pulleys on a jack- 
shaft is liable to produce a greater bending moment, it is 
customary to assume a larger shaft to a given 


horse-power; therefore, instead of using —— as the working 

horse-power transmissible by the shaft, it is better to use 

under these conditions HP = -— — . 


From this value of the power transmitted we obtain 

The ratio of power absorbed by friction to the horse- 
power which the shaft is capable of safely transmitting will 

now become 

HP, O.O'SdNF O.IOF ^ ,,^, 

HP = 00085^ = -W- ^'' ^^^^' • ' ^^^) 
from which Table XVII has been determined. In calcu- 
lating the values given in this table it has been assumed 
that the belt speeds are the same as those previously con- 
sidered, namely, 2640 feet per minute for main belt to 
head shaft, and one fourth of this, or 660 feet per minute, 
from head shaft to auxiliary shafting. This may be low in 
many cases, but, as already pointed out, the force F will 
decrease under the assumed conditions as the belt speed 
increases, so that we may expect the friction loss obtained 
by the above formula to be somewhat less than the actual. 
In the foregoing discussion it has been assumed that the 
shaft transmits its allowable maximum power, that is, for 

line-shafting the power transmitted = txtt, and for head- 

shafts the power transmitted =— — . 

As a general thing, the actual average power transmitted 
by a shaft is not more than about three-fourths of its as- 
sumed working capacity; and since the weight and speed 
remain practically constant, the percentage of loss under 
conditions approximating those we have assumed will be 
somewhat greater than that given in the table. But while 



the power transmitted may be diminislied 25 per cent, the 
percentage of increase in friction loss will vary between wide 
limits, depending upon the speed of rotation and length of 
shaft. Thus for a 3" head-shaft 100 feet long, delivering 
three-fourths of its allowable capacity at 100 revolutions 
per minute, the loss increases from 9.0 to 11.5 per cent, 
which represents a gain of 28 per cent; at 250 revolutions 
the loss is now 14.0 per cent, corresponding to an increase 
of 14.3 per cent; while at 400 revolutions the loss is 17.8 
per cent — a gain of only 10.6 per cent. 

Table XVII. — Power absorbed by Friction in Jack or 

Head Shafts. 

Diameter of 


Percentage of Loss when Length iu Feet — 

in Tnclies. 














( 100 










( 100 






























For these determinations it is supposed that the shafting 
is properly supported, with hangers suflBciently 'close to 
each other to prevent undue deflection under working con- 
ditions, and that the shaft is in line, having good bearings 
lubricated us in common practice.* Departures from these 

*In the iibove discussion it was assumed that the coefficient of 
friction is constant and that the friction varies directly as the load. 
While recent experiments on machine journals running iu oil indi- 
cate that the coefficient of friction varies inversely with the load 
there seems no good reason lo doubt the truth of Moriu's laws for 


assumptions will further increase the friction loss; but, 
on the other hand, this loss will be decreased if lighter or 
fewer pulleys be used throughout the length of the shaft, 
if the bearings be continuously lubricated, or if the ma- 
chines be belted directly from the shaft below. Where 
shafting is employed there will generally be an additional 
loss due to the friction of the auxiliary shafting and coun- 
ter-shafts, which is extremely variable. 

It is outside the province of the present subject to discuss 
the losses in these secondary shafts: the losses which we 
have here been considering are those which exist in main 
line-shafting, jack- and head-shafts receiving their power 
presumably by leather-belting or ropes, with either of which 
for similar drives there should be no appreciable difference 
in the total weights of pulleys and shafting and the friction 

The diameter of grooved shaft-pulleys will be larger and 
the rim will be thicker; but for the same horse-power trans- 
mitted the width will be less and the weight not materially 
increased. In any case the total weight of grooved pul- 
leys compared with that of belt-pulleys used in the same 
system is very small, and any individual differences may 
be neglected when taken as a whole. In those cases where 
ropes are used exclusively, as, for instance, in dynamo rooms 
and other power-stations, the pulleys are frequently heavier, 
and the shafting usually is fitted with a number of friction- 
clutches, thus materially increasing the weight on the bear- 
ings; in cotton-mills also, where the ropes are geared di- 
rect from the engine to the various floors of the mill, there 
is frequently a heavy stress on the shaft, especially on the 
upper floors, due to the weight of the ropes: under such 

such comparatively rough and imperfectly lubricated bearings as we 
have been considering in which the friction between the rubbing 
surfaces in contact and not the viscosity of Ihelubricnnl is a mca'^ure 
of the resistance. See paper by Prof. Denton in American Machinist^ 
Oct. 33. 1890. 



conditions the shafting should be considered as head-shafts. 
In work of this nature the velocity of the ropes is usually 
much greater both from the engine to the first shaft, and 
from the latter to the machine or secondary shaft. Assum- 
ing a speed of rope double that used for the previous tables, 
and letting the weight of pulleys, ropes, clutches, and coup- 
lings equal four times the weight of shaft, it can be shown 
that the formula for the friction loud will be reduced to 

F= V[OM'Lx 0.0057^'iV]'' + (0.mUN)\ 

Table XVIII.— Power absorbed by Head-shafts carrying 

High-speed Ropes. 

Diameter of 


Percentage of Loss when Length of 
Shaft in Feet = 

Shaft in inches. 

per Minute. 











































Line of action of resultant of tensions in main drive = 30°. 
Velocity of ropes from engine 5280 feet per minute. 
Ropes from shaft are horizontal, and run at an average of 2640 feet 
per minute. 

All ropes drive from one side of the shaft toward the engine. 
Weight on bearings four times weight of bare shaft. 

From tills formula Table XVIII has been calculated, 
and may be considered to represent the percentage of loss 
in the fii'st shaft when working under full load; with 
lighter loads the percentage will be greater. 

It must be noted that with any other arrangement of shafts 
the friction loss will vary. In the present case the ropes 
from the engine to the jack-shaft make an angle of 35° with 


the horizontal, and the ropes from the jack-shaft back to 
the machine or secondary shaft are horizontal, as in Fig. 54. 

When rope- wells are used, each successively higher shaft 
will have an increased friction percentage, since the belt- 
pull becomes more nearly vertical, and the resultant load 
on each shaft is thereby increased. 

It is worthy of remark that in long lines of shafting 
with high rim velocity the influence of belt-pull on the 
bearings is very slight compared to the weight of shaft 
and pulleys, so that the loss in friction is but little more 
than that due to weight alone. We see in this an addi- 
tional argument for high rotative speeds in shafting, for, 
while the percentage of loss increases, from 8.1 to 9.2 in a 
2i-inch shaft 100 feet long running at 100 and 400 revolu- 
tions per minute respectively, the power transmitted by the 

shaft, as calculated from HP = -r^— - increases from 21^ for 


100 revolutions per minute to about 87 h. p. for 400 revolu- 
tions per minute, so that while the friction percentage 

9 2 
increases in the ratio . ^ = 1.14, the power transmitted 


increases in the ratio of the number of revolutions per 

minute, or 4 to 1. In the first case the friction loss is 

21^ X .081 = 1.74 h. p.; and in the second, the loss is 

87 X .092 = 8.00; therefore the net power transmitted by 

the shaft running at 100 revolutions per minute is 19 h. p. 

whereas by increasing the speed to 400 revolutions per 

minute the net power transmitted will be 79 h. p. With 

higher belt velocities and increased rotative speeds in our 

factory shafting the friction loss, instead of being from 30 

to 50 per cent of the total power transmitted, ought not to 

exceed one-half of these percentages; for with higher speeds 

narrower and lighter pulleys could be used, the belts could 

be run slacker, and lighter shafting could be employed. 

Although it has been previously considered in a general 


way that the friction due to the sliafting may be taken as 
about 10 per cent of the full load transmitted to tlie shaft, 
yet, in the light of further investigation, it will be seen 
that, owing to the various conditions under which the 
shafting is run no general value for friction loss can be 

While the friction absorbed by large engines is reason- 
ably less for rope-geared fly-wheels when compared with 
engines using flat belts, driving in a-similar manner, there 
seems to be no good reason for supposing tliat the friction 
in ordinary mill-shafting should be appreciably different 
when rope-driven. 

On account of the larger diameter of pulleys used with 
rope-driving, the velocity of the rope may be, and usually 
is, greater than that in a belt used in the same place, and 
for this reason the pull on the shaft due to the tensions in 
the rope maybe less; but with long lines of rope-driven 
mill-shafting the main drives only are of rope, and any 
difference in pull on the bearings which might exist in 
favor of the rope-driven main shaft must necessarily be 
small when compared with the total friction load due to 
the pull of the numerous cross or machine belts which, 
running at a greatly reduced speed, produce by far the 
greater effect Sn the shaft. With short lines of shafting, 
Jiowever, there will generally be a small saving in favor of 
a rope-driven plant. Under these conditions the effect of 
the ropes to and from the first motion shaft is usually in 
excess of that due to the belts which may be used to trans- 
mit power from the main or jack shaft to secondary shafts 
or machines; and therefore, since the rope-pull is less than 
would be produced by belts used in the same place, we 
may expect the friction to be less. When ropes are used 
entirely, as in electric and other power stations, we should 
expect the friction loss to be somewhat less, assuming that 
the ropes are run at ;i higlier speed than would be used 
• belts in the same place. 

B.0PE-DBIV1NG, 159 


It has been stated that a further decrease of from 5 to 
8 per cent of the power transmitted by a rope may bo at- 
tributed to losses in the rope itself due to resistance to 
bending, wedging in the groove, differential driving effect, 
and creep, all of which affect the loss to a greater or lesser 

Various formulas have been proposed by several eminent 
authorities by which the resistance of a rope to bending 
might be determined. Eytelwein's formula assumes that 
the resistance of a rope is directly proportional to the ten- 
sion and the square of the diameter, and inversely propor- 
tional to the radius of curvature of the pulley; in which 
case the stiffness of a hemp rope for each v^inding and un- 
winding is given by 

a = c—T. 

where c is a constant equal to 23, d is the diameter of 
rope in inches, r is the radius of pulley in inches, and 7' 
is the tension in the rope. If the ratio of the diameter of 
rope to the diameter of pulley over which it runs equals 1 
to 30, the above formula becomes, for a rope running over 
two pulleys, 

(T = 0.03^!r. 

Reuleaux states that, since transmission-ropes are usu- 
ally quite slack, the coefficient of stiffness should be taken 
somewhat less than Eytelwein's value, and suggests that 


two-thirds would represent a fair approximation; this 
would give 

cr = ^X 0.23-7=0.15-7 
3 r r 

for each pulley in the system. 

If the tension on the tight and slack sides of the rope be 
represented by 7, and 'l\y respectively, the average load 
on the rope may be considered equal to ^(7\-|-T,); if, 
further, the conditions be assumed such that the slack-side 
tension equals one half that in the tight or driving side, — 
we shall have 7'= i(t7,). Hence for two pulleys, when the 
diameter of the latter equals 30 times the diameter of rope, 

^ = Vl55-X2X2^«) 
= 0.02d X i7\. 

Since 7\ has been taken in our previous work as equal 
to 2006^' pounds, the stiffness in the rope will now become 

(T = 0.02d X 150(i* 
= U\ 

Now under these relations of tension the driving force 
may be obtained from 

P^T,'-T^:= 200^* - ^- = lOOcP; 
lience the ratio of loss due to bending will be 

F "" \00(P' 

For a f-inch rope running over two pulleys the loss 
equals 2.25 per cent, while for a 2-inch rope under similar 
conditions the loss becomes 6 per cent. This is wluit wo 


might expect; for it is reasonable to suppose that the per- 
centage of loss should increase with the diameter of rope. 

It will be noticed that the work done in bending a rope 
over its pulley is directly proportional to the number of 
bends, and therefore in designing a rope transmission 
every effort should be made to restrict the number of 
bends, as this is not only a large factor in the wear, but, as 
just shown, the power transmitted for a given tension is 
constantly reduced as the number of bends increases. 

This feature is a decided disadvantage with installations 
on the continuous-rope system where one rope is bent in 
both directions around a number of pulleys on the several 
floors of a factory. Under such conditions, and also in those 
special cases where it is absolutely necessary to run pulleys 
smaller than that obtained from the formula* D=d}''^ X 

I^F+12", the usiBr should put in the very best quality 
of loosely twisted rope and run it at a less strain than 
would otherwise be adopted ; for under such conditions the 
flexibility and elasticity of the rope are more desirable than 
a high breaking strength. 

While the foregoing formula of Eytelwein may give a 
measure of the force required to bend a rope over a 
pulley under a certain set of conditions, it will be 
evident, since the conditions vary considerably in dif- 
ferent installations, that, in order to be generally appli- 
cable to any given case, a formula must contain other 
factors than those included in Eytelwein^s and other simi- 
lar formulsB. In a flying rope running in V grooves, in 
addition to the bending of the fibres there is a permanent 
reduction in cross-section, due to the uniform compression 
of the rope, as will be noticed by measuring a rope that 
has been running some time; besides this there is a tem- 
porary deformation due to the distortion of the rope as it 
passes over the pulley. 

* See page 179. 


The flexibility and elasticity of the rope undoubtedly 
have much to do with the resistance to bending, and the 
degree of twist put into a rope is an important factor in 
this connection; for although hard -twisted ropes are very 
much stronger than those loosely twisted, the internal 
wear is much greater, as the fibres are held more rigidly 
and do not slide as freely upon each other. The advan- 
tages of lubricating the fibres of a man ilia rope have been 
already discussed; it is sufficient to state here that the 
degree of lubrication affects the flexibility of the rope, and 
hence enters as a factor in determining the loss due to 
bending. The varying angle of contact and, to a lesser 
extent, the angle of the groove, must also have a certain 
influence upon the resistance. In view of these considera- 
tions it will be seen that any deductions from existing for- 
mulae are of doubtful utility when applied to transmission- 
ropes in use, and shoulcj be considered only as relative, and 
not absolute. 

Another source of loss is that due to the wedging of 
the rope in the groove. Although this action exists to a 
greater or less extent in all rope transmissions where the 
shape of the groove is such that the rope does not bottom, 
yet it is undoubtedly true that its effect in a well -con- 
structed plant has generally been over-estimated. That it 
does exist to a harmful degree can be seen in many instnl- 
lations from the way in which the tight side of the rope 
follows upon the driven pulley. With the single-rope 
method this is especially true in new installations, where 
the tension is purposely made rather high to allow for 
stretch in the rope; it is, however, frequently found in the 
continuous-rope system where the tension-carriage is over- 
weighted. The factors which enter into the consideration 
of this loss are: Tension on tli« driving and slack sides of 
the rope, the angle of groove, and the velocity, weight, 
and condition of the ropes. As we have previously shown 


(page 132), the driving-side tension T^ is made tip of three 
parts, namely, the driving force P, the centrifugal force 
F^y and the tension T,, necessary to balance the strain for 
adhesion; that is^ 

T, = P + F,-\-T,. 

In like manner the tension in the slack side of the rope 
consists of the strain necessary for adhesion plus the 
strain due to centrifugal force, that is^ 

It has been claimed that no loss can occur in pulling the 
rope out of the groove, since the centrifugal force set up 
in the rope is many times greater than any caused by the 
tension on the slack side when leaving the pulley; but it 
is obvious that a part cannot be greater than the whole, 
and therefore the centrifugal foi-ce, while greatly reducing 
the wedging force, cannot altogether eliminate it. 

An abnormal degree of slack-side tension has a direct 
effect upon the wedging of the rope in the groove, for if 
sufficient sag is not allowed and the slack-side tension is, 
therefore, needlessly great, it follows that the power trans- 
mitted will be reduced, since the driving force, P, is equal 
to the difference of the tension in the two portions of the 
rope = T^— T^\ on the other hand, the force drawing the 
rope into the groove will be increased, since the force 

r, - ^, + r. - i^„* ^ p + t; + r, 

= p + 2r,. 

It is obvious, therefore, that the slack -side tension 

* Although the centrifugal force increases the tension in the rope, 
its tendency is to cause the latter to leave the pulley, and therefore 
it should not be considered as a part of the forces drawing the rope 
into the groove. 


should be no greater tlian just sufficient to give adhesion 
to the ropes and prevent undue slip at the desired speed. 

As to the best angle for the groove in the pulley, opinion 
is still somewhat divided; but in England the general prac- 
tice seems to favor an angle of 45° as the most suitable.* 

In the earlier installations a more acute angle was used, 
7»tj{.. ^ as evidenced by the discussion of Mr. Durie's paper on 
Rope-driving, presented to the British Institution of Me- 
chanical Engineers.! Grooves having an angle of 30° had 
been tried, but it was found that the wear on the rope was 
altogether too great; 40° was a very satisfactory angle, and 
is still preferred by many engineers, while others use as 
large an angle as G0°. Occasionally half-round grooves 
are used; but with semicircular grooves on cast-iron 
pulleys either the tension in the rope must be increased 
or a greater number of ropes must be employed: in any 
case the advantage seems doubtful. With wooden-rimmed 
pulleys, however, the semicircular groove is the better 
form; for since the coefficient of friction on the wooden 
pulley is from thirty to fifty per cent greater than for a 
similarly shaped groove on an iron pulley, it follows that 
the tension in the rope need not be so great; moreover, 
* wooden pulleys that have been in use for some time would 
indicate that the semicircular groove is better adapted to 
the work, for with anything but very light loads the angu- 
lar groove is soon cut out by the rope, producing a rim 
somewhat similar to the one shown in Fig. 57, which rep- 
resents an angular-grooved wooden pulley that had been in 
use but a few months. 

AVith a semicircular groove in the first place the latter 
will retain more nearly its original form, and the wear on 
the rope will be greatly reduced. 

Tiie pliability of the rope has considerable influence on 

* M. E. iu American Machinist, Nov. 10, 1892. 
f SeeProc. Inst. M. E. 1876. 


the shape of the groove; while a 30° angle may give the 
correct shape for a soft, loosely twiated cottoii-rope, a harder 
twist may reqnire an angle of 40° or even 50°; in the same 
way a 40° groove may be nil right foi some makes of ma- 
nilla rope, while others of a less yielding nature would give 
better results if an angle of 50° or even GO" were used. 

At the present time there are very few pulleys nsed in 
this country, except for machine bands, liaviug grooves 
with an angle less than 40°. Formerly the Yale & Towne 

Pig. 57. — Btst of Wooden Pulley Showing Wear. 

Mfg. Co. employed grooves of 30° for small cotton ropes 
{about 3 inch in diameter) driving their ti-avelling-cranes; 
but these were subsequently changed to 50°, and at the 
same time larger ropes, IJ inch, were adopted. 

For manilla rope-drives one large manufacturer uses an 
angle of 60° on all his pulleys, whether of wood or iron; 
this angle was arrived at after much trial, and represents an 
experience with manilla-rope transmiBsion, covering a great 
many years. 

The most suitable angle of groove is that which affords 
the greatest frictionul adhesion without undue slip, and at 
the same time offers the least resistance to the rope in leav- 
ing the groove: with much slip the rope is rapidly worn 
out, while with an excessive grip the wear is also rapid, and 
a relatively large amount of force is absorbed in overcoming 
the wedging. The usual practice in this country for both 
cotton and manilla ropes is an angle of 45°; in many cases 



the section of the groove is formed by area of circles, liav- 
ing a r^iUB equal to from 3 to 4 diainetora of rope, in wliicli 
case the included anglu is consbtntly chuuglng, and the co- 
efficient of friction, and lienee tli<! grip, will vary with the 

diameter of rope used. Thui 

in Fig. 58, if AB and A'B' 

represent two 
,j9 „■-'' ropes of aliglitly 
'--'' different diame- 

ters running over 
a grooved pulley, 
the one which 
sinks deeper into 
the groove will 
\ i include the 

Vc greater angle of 

'' p KB contact ^£C,and 

hence the lesser 
coefficientof friction. This will hold true of two ropes hav- 
ing the same diameter but different degrees of twist: the 
harder rope will not sink as deeply into tlie groove, and its 
coefficient of friction will be greater than that of the smaller 
rope, other things being equal, on account of the leaser 
angle in the groove at the point of tangency of the harder 

With the excellent ropea that are now being made for 
transmiasion purposes this form of groove possesses many 
advantiigea, even with the continuous-wound system. 

In tliis aystem the tension is assumed to be the same in 
each wrap; hut there is unquestionably a variation of pnll 
due to momentary fluctuations, which must either be ab- 
sorbed by the elasticity of the rope or ti'ansmitted through 
the rope until the strains are equalized. If the rope used 
is uniform in structure, that portion which receives the 
greater strain will be drawn more deeply into tlie groove; 
and if the latter he of the curved form, the coefficient of 

ROPE-DRtVtKG. 16'/ 

friction will be reduced: so that the resultant adhesion pro- 
duced in this portion of the rope will be less than that in 
some other part which, under a lighter strain, occupies a 
position on tlje pulley such that its coefficient of friction, 
and consequent adhesion, is greater for a given back-tension. 
With the usual arrangement of pulleys, namely, that in 
which the larger wheel is the driver, the tendency of an 
increase in tension is to increase the velocity of the driven 
pulley. If several wraps of a series occupy positions in the 
grooves such that with a given ratio of pulley diameters 
the velocity of the smaller pulley is twice that of the driver, 
any decrement, a:, of the effective radii of another wrap, 
which is drawn more deeply into the grooves of the two 

pulleys, will alter the velocity ratio from - to :j — , a quan- 

tity which must of necessity be greater than 2. The ten- 
dency of this wrap then is to produce a greater velocity in 
the driven pulley, which cannot occur without some slip in 
the other wraps; but these wraps have a greater adhesion, 
and therefore tend to drive the pulley at a less number of 
turns per minute, which will produce slip in the more 
heavily strained member: hence the effect of any change 
in position due to sudden increased strain on one wrap 
will tend to quickly adjust the tensions in all positions of 
the rope and neutralize any inequalities in driving effort. 
This form of groove, as we shall show subsequently, is even 
more desirable for the individual rope system, where the 
evils of differential driving are frequently so pronounced. 
As noted previously, the frictional grip depends both 
upon the arc of contact of the rope with the pulley and 
the coefficient of friction, which latter varies with the angle 
of the groove. In order, therefore, to produce the same 
friction on each pulley, the product of the arcs of contact 
by their respective coefficients of friction must be equal. 
If, as before, we take the coefficient of friction of a well- 

168 jaoi»E-DiiivmG. 

lubricated rope on a smooth, flat metal pulley equal to 0.12, 
the coefl&cient for the same rope in a groove whose angle 

is 6 degrees will be = 0.12 cosec— ; hence with arcs of 


contact a and or' we should have for an equal grip on each 


(pa = (p'oc'j 

0.12 cosec - X a = 0. 12 cosec — X a', 

assuming that the multiplier 0.12 will give the correct co- 
efl&cient of friction on each pulley, since the percentage of 
slip is the same. 

As the numerical value of the cosecant of an angle varies 
inversely with the angle, it will be obvious that the pulley, 
having the lesser arc of contact, should also have the more 
acute angle in the groove. This property of rope friction 
is frequently taken advantage of in. designing a plant, and 
it is not uncommon to find the grooves in the large wheel 
more obtuse than those in the smaller, especially when 
there is considerable difference in the diameters of the 

From the above equation there is obtained 

6 6' a' 

cosec — = cosec -TT X—, 
2 2 a 

in which and a represent the angle of groove and arc of 
contact, respectively, on the larger pulley, and 0' and a' 
similar values for the smaller pulley. With the least angle 
of groove equal to 35°, 40°, or 45°, the corresponding angle 
in the larger pulley should be as indicated in Table XIX, 
when the ratio of the arcs of contact is known. 

*Mr. T. Spencer Miller, of tlie Lidgerwood Mfg. Co., has been 
granted a patent (U. S. patent No. 444919), upon the application of 
this principle to continuous- rope transmissions, in which with Sheaves 
of different diameters more obtuse grooves are given to a larger wheel. 



Table XIX. — Angle of Groove for Equal Adhesion. 

Arc of contact on small pulley _ a' 


Arc of contact on large 
Angle of groove in large 

when groove in small 
Angle of groove in large 

when groove in small 
Angle of groove in large 

when groove in small 

pulley -35' 
pulley =40' 
pulley = 45' 
























Of course an idler or binder pulley may be used to in- 
crease the arc of contact on the smaller pulley, and thus 
maintain an equality of grip on each pulley; but this device 
produces an objectionable reverse bend in the rope, which 
should be avoided as much as possible in rope transmission. 
Both of these arrangements, as well as the winder-pulley 
previously discussed, are intended to prevent slip and at 
the same time obtain the maximum adhesion for the least 
amount of back-tension without increasing the losses due 
to wedging in the grooves, journal-friction, and wear in 
the rope. 

As pointed out by Mr. W. H. Booth,* although there 
may be some loss and wear due to wedging in the groove, 
by far the greater loss is occasioned by the deeper wedging 
of one rope as compared with another, so causing them to 
grip upon a different circumference, in which case each 
rope tends to impart a different velocity to the driven pulley; 
the actual resultant velocity will be a mean of the several 
velocities of the individual ropes, so that slipping and wear 
of some or all of the ropes must occur, due to the differen- 
tial driving thus set up. 

AVith continuous rope transmissions this effect is not 
so apparent, although it exists to a certain extent on account 
of inequalities in the rope and various mechanical imper- 
fections in the system; but in the English or independent 

* American Machinist, Dec. 8, 1888. 


rope traDsmission its effect is very marked, and is generally 
considered as the principal source of loss of efficiency in 
this system.* 

While there is undoubtedly a considerable loss which 
may be charged to this cause, it is also true that its effect 
may be greatly reduced by a careful study of the require- 
ments of the problem and an intelligent application of 
correct principles to the case in hand. 

It is evident that in order to prevent slip and the loss of 
power incident thereto it will be necessary to obtain a uni- 
form velocity in the several ropes running over a pair of 
pulleys; to approach this desideratum each set of ropes 
should be of the same make and degree of hardness, of 
uniform diameter, evenly spliced, having the same amount 
of sag in both members, and run over grooves of uniform 
diameter, shape, and smoothness. 

Obviously it would be impossible in practice to maintain 
all these conditions, even if it were practicable to overcome 
the mechanical difficulties and install a plant under the 
given requirements ; yet much may be done to effect the 
desired end. 

Uniformity of length is an important requirement, for, if 
one rope of a set be allowed a less amount of sag than the 
others, the sum of the tensions will be greater in this rope, 
and in consequence it will be drawn more deeply into the 
groove; its pitch diameter will therefore be reduced and 
its velocity will be different from the others in the set, in 
which case if the driving and driven pulleys are of unequal 
diameters the tendency will be to give the driven pulley a 
different velocity, and slip must necessarily occur. 

The following from the American Machinist is pertinent 
to the subject: 

" It will be readily seen that in a set of ten or a dozen 

* Amei'ican Machinist ^ Dec. 1, 1898. 


ropes, each of a somewhat different length, the loss of 
power from this cause may easily become a very serious 
item, and to this there is to be added the correspondingly 
diminished life of the ropes. A similar action occurs 
when worn ropes are allowed to work conjointly with new, 
even though the deflections, and therefore the tensions, on 
the several ropes are practically equal. In this case the 
loss due to abnormal tension and wedging of some of the 
ropes into the grooves is avoided, but tlie differential driving 
effect due to the ropes virtually running on pulleys of 
different diameters still exists, and is equally objectionable. 
It may here be noted that the effect of the differential 
driving upon the ropes depends to a very large extent 
upon the relative diameter of the two pulleys. 

" It is evident that, when the driving and driven pulleys 
are of the same diameter, any variation in the effective 
pitch diameters of the several grooves will have no appre- 
ciable effect upon the transmission, provided that the 
diameter and shape of the corresponding grooves in the two 
pulleys are the same. It may be noted, however, that the 
worn ropes which run deeper in the grooves, having a 
slightly less velocity, are subjected to a somewhat greater 
stress than their newer and larger companions." 

With a form of groove similar to that shown in Fig. 58, 
in which the sides of the groove are circular arcs of large 
radius, there will be a tendency to correct the differential 
driving, especially so when the driving pulley is smaller 
than the driven. In this case, if we assume that all the 
ropes in a set are put on with the same amount of sag, so 
that the tensions are practically equal for both old and new 
ropes, any difference in diameter of rope will cause the 
larger to carry more than its share of the load, since the 
effective radius of the pulley, and therefore the velocity of 
the rope, is greater; moreover, since the coefficient of fric- 
tion varies inversely with the depth of contact in this form 


of groove, the larger rope is acted upon by a more intense 
grip, by reason of which more work is imposed upon it, 
while a smaller rope will be relieved of some of its work. 

The tendency of the larger rope is to turn the driven 
pulley at an increased velocity. . Thus in Fig. 59 if D be 

\\pmW5i " \\ j i 

^^^^^rrzz A roLLowER // 

R— "f F EUe. World 

Fig. 59. 

the driviug-pulley and i^the follower, with respective diam- 


eters such that the velocity ratio — is 1 to 3, then the 

smaller ropes will tend to drive i^in the ratio J, while 
the larger ropes, n working at a distance x further away 
from the centre of pulleys, will tend to drive F in the ratio 

, and therefore at a higher velocity. The resulting 

velocity of the follower will depend upon the work done by 
each rope, so that some slip must necessarily occur; on 
account of the greater load taken by the larger ropes they 
will be rapidly compressed and worn: hence any initial 
variation in turning effort will be speedily reduced to a 
minimum by this equalizing process, which, although it 
incurs loss, ultimately insures a better distribution of the 
work with the least wear on the ropes. 

On the other hand, when the driving-pulley is larger 
than the follower the smaller ropes are drawn further into 
the grooves and tend to impart an increased velocity to the 
driven sheave. Thus in Fig. 60 the large pulley is the 

R 3 
driver and the velocity ratio is — = - with the smaller 

r 1 


ropes o; with the large ropes n, workiDg at a distances; 
nearer the circumference, the tendency will be to produce 

a ratio equal to :j : hence the effect will be retardation. 

Since the larger ropes acting higher in the groove have an 
increased grip and speed they will exert a greater influence 
upon the smaller pulley; and although the smaller ropes 
may be drawn more deeply into their grooves in attempt- 



Fig. 60. 

ing to drive the follower at a greater number of revolutions 
per minute their effect is lessened on account of the dimin- 
ished grip, due to the form of groove, in consequence of 
which the larger rope acts with a greater effect and the 
smaller with a lesser effect than would obtain with an ordi- 
nary V-groove under like conditions. 

To equalize the driving efforts of a number of ropes, and 
to prevent the slip which must inevitably occur with a 
solid-grooved rim, Mr. John Walker has devised and pa- 
tented * a " differential " driving-pulley, since it allows the 
ropes to travel at different speeds suited to the conditions 
imposed upon each rope. 

Originally intended for cable-railway machinery, where 
the wear on the drums due to the wire cable is excessive, 
the differential principle has been extended to other uses, 
notably elevator sheaves and rope-transmission pulleys. 

In the latter the rope is led over a number of separate 

* Feb. 23, 1893. 


rings. Fig. 61, adapted to turn loosely and independently 
of each other on the 
smooth circumfer- 
ence of the drum. 

While the rope is 
pasaingover the pul- 
ley the tendency of 
the rings will be to 
adjust themselves to 
the strain in each 
member by moving 
around the circumference of the drum. Thus the driving- 
tension IS equalized, ind each rope is brought to do its own 
share of the work without slipping in its groove. These 
rings have a diameti ical friction, due to the pressure of the 
rope in the groove transferred to the flat surface of the 
drum. In addition to this an adjustable rubber washer is 
inserted between the rim of pulley and loose flange, so that 
by tightening this adjustable joint the separate rings are 
caused to exert enough pressure upon each other to pro- 
duce a certain amount of friction on the side surfaces, the 
combined friction of the several rings being sufficient to 
drive the pulley or ropes, as the case may be. In this way 
each rope bears its due share of the work, as the adjust- 
ment is such that the friction between the several parts 
is brought into equilibrium. 

In practice the axial rotation of the ropes will frequently 
exert a modifying influence on the differential driving, 
since, other conditions being unchanged, a rotating rope 
tends to maintain its circular form, and therefore will work 
loss deeply into the groove. An additional advantage is 
that such rotation promotes the durability of the rope, as 
the wear is more uniform. 

The loss due to the elastic slip or creep of the belt hag 
some influence upon the efficiency of tnmemission, but in 


any case its effect is small. When an elastic body, such 
as a rope, is placed under tension, it stretches, and the 
elongation, within the limit of elasticity, is proportional to 
the strain in the rope. When power is transmitted from 
one pulley to another the driving side is subjected to a 
greater tension than the slack side, in consequence of which 
the velocity in the driving side will be slightly greater than 
that in the slack side, and the circumferential velocities of 
the two pulleys will not be the same. This will be evi- 
denced from the following considerations: 
Let V = circumferential velocity of driver; 

F' = " " " follower; 

T^ z=z tension in driving side of rope; 

y, = " " slack side of rope; 

A = cross-section of rope in square inches; 

L = original length of a piece of the rope in either 
= side in its normal condition; 

e = elongation in driving side due to tension T^ ; 

e' = *' " slack side due to tension T^; 

E = modulus of elasticity of the rope. 

T T 

When the rope is at work e = -=-; a"d e' = -=-?, so that 

JifA Ji/A 

the length of each member will now he L-\-e and L -\- e', 
respectively. The length of rope running on to the driving- 
pulley in a unit of time will therefore be greater than that 

L 4- e 
delivered to the driven pulley in the proportion ,. „ and 

the velocity of the two pulleys will now be 

^-^i-- hence F-Fx ^ + ^' 
r-L + e'' nence K - KX-^q^^. 

T T 

Calling L unity, and assuming -7=^* = 2, -^ = 320 pounds 

V, A 

(since T^ = 200^? pounds, and A = 0.8x~d^), and ^ = 


40000,* we have 

320 - , 320 

40000 * ^ '^ 2X40000' 

r=Fx— «^« = 0.996F; 

that is, the loss due to creep when all the ropes are working 
under normal conditions is about one half of one per cent. 
With different ratios of driving to tight-side tension and 
different intensities of stress it is evident that this loss will 
vary in the different ropes; in any case the loss ought not 
to exceed one and a half per cent, as will be seen if we as- 
sume extreme conditions. 

For instance, let the stress on the rope be 600 pounds 
per square inch of section and assume the ratio of tensions 
equals one to five; with the most elastic long-staple cotton 
rope it is possible that the modulus of elasticity may be re- 
duced to 30,000 pounds; therefore, under these conditions, 
the velocity of the circumference of the driven pulley will 

14- -^^^— 

■^■^ 30000 

thus representing a loss of 1.6 per cent. It is probable 
that one per cent is an ample allowance, even under un- 
favorable conditions. 

* See Reports of Tests on ManUla Rope, Ex. Doc. No. 36, 1885. 



The construction of rope pulleys is a matter of consider- 
able importance, for it is evident from what has preceded 
that the size, shape, and condition of the pulley all exert a 
marked influence upon the efficiency of rope transmission. 

It has been shown that the size of the pulley materially 
affects the wear of the rope: the larger the sheaves the less 
the fibres of the rope are flexed, and the less they slide on 
each other; consequently there is less internal wear of the 

The minimum diameter of the pulley, as given by differ- 
ent authorities, varies from thirty to forty times the diame- 
ter of rope to be used. Forty times the diameter of rope 
for manilla is excellent practice, and experience has shown 
that a larger multiplier would be still better, as the larger a 
pulley, the better for either belt or rope passing over it; 
but such a rule, although .convenient to use, was evidently 
founded upon large-sized ropes running at a high velocity, 
and a little consideration will show that while forty times 
the diameter of rope may be all right for a two-inch rope, 
it is also true that a lesser proportion will give suitable di- 
ameters of pulley for a one inch rope. If we take two ropes 
rf, and d^y whose diameters are one and two inches, respec- 
tively, and bend them over pulleys of the same diameter, 
the fibres in d, will be stressed an amount equal to x due to 
the stretch and sliding of the fibres one upon another. On 
the other hand, since the outer fibres in d^ are twice as far 
from the neutral axis, tliese fibres will stretch and slide 
upon each other to an extent equal to twice that produced 


in the smaller rope ; bat the relation of the areas of the 
two ropes is 

under the given conditions ; hence the total slip in the 

larger rope, due to the sliding of the fibres upon each other, 

will be eight times greater than in the smaller one, or as 


-j^. Therefore, for equal extension and slip of the fibres in 

bending a rope over a stationary pulley, it would appear 
that the pulleys should have diameters proportional to the 
cube of the diameters of the ropes. 

In the above consideration no account has been taken of 
the external wear, produced by slipping in the groove, 
wedging, rubbing contact, and other causes of loss which 
affect a running rope; for a given velocity these losses will 
increase with an increased speed of rotation, and hencje will 
be greater with a smaller pulley, which would indicate that 
the diameter of the latter should not be directly propor- 
tional to the cube of the diameter of the rope. It is also 
evident that a rope running at 5000 feet per minute will 
be subjected to a greater number of bends and a greater 
external wear than it would if running over the same pul- 
leys at 2000 feet per minute. The first of these influences 
is recognized by many engineers who use a rule approxi- 
mating the following: 

For the least diameter of pulley, D, multiply the circum- 
ference of rope by ten times its diameter and divide by 
two. * This is practically equal to D = 16d^ inches. 

While we believe in using as large a pulley as possible in 

*Mr. Jas. Gamble in Textile Recorder, 


any given case, conditions will arise when it is desirable to 
use the smallest possible diameter without excessive injury 
to the rope. In such case the individual circumstances 
should be considered by the designer. 

From the foregoing it is obvious that for a given rope 
and tension, the least diameter of pulley which may satis- 
factorily be used under known conditions should be de- 
pendent upon the velocity of the rope, and should vary 
with the size of the latter in such a manner that for any 
given speed the pulley diameter will be proportional to 
some power, greater than unity and less than the cube of 
the diameter of the rope. 

From an investigation of numerous examples in operation 
under varying conditions, some of which work satisfactorily 
and others very poorly, it would seem that while the value 
ISeT may give a suitable diameter of pulley for a soft cotton 
rope of small diameter, such a value is entirely too small 
for an equal-sized manillarope; but, on the other hand, 
40d is somewhat larger than absolutely necessary for these 
ropes, although it is always desirable to use such a pulley 
if conditions permit. 

If for any reason it is necessary to adopt a small pulley, 
the least pitch diameter for a sheave to be used with 
manilla rope working under an assumed tension of 200^/" 
pounds may be determined from the following empirical 
formula, which is believed to represent the requirements of 
good practice: 

D = d'-' X t^r+ 12", 

in which D = pitch diameter of pulley in inches; 
d = diameter of rope in inches; 
F= velocity of rope in feet per minute. 
In order to simplify the use of this formula the follow- 
ing values of d^"^ and t^ V have been calculated as given 
in Table XX. 



Table XXI gives values of D for ropes varying from f 
iuch to 2 inches in diameter, running at 2000 to 5000 feet 
per minute. When the speed of the rope is not known the 
diameter given in the last column should be used for the 
minimum size of pulley; in fact it would be better to use 
these diameters, or even larger ones, in all cases, provided 
the constructive features in the plant will permit their ap- 
plication. With an increased tension in the ropes the diam- 
eter of pulley should be increased also. In the same way 
if the working tension should be less than 200^?' pounds, 
then the diameter of pulley may be less than here given. 
If cotton rope be used, the least diameter of pulley may 
also be taken somewhat less. 

Table XX.— Values of (?•' and ^V. 

Dia. of rope d | 

Viilueof eP-7 

Velocity of rope, F, in feet 
per minute 

Value of VV 























Table XXI, — Least Diameter op Pulley for Given Diame- 
ter AND Speed of Manilla Rope. 

D = (rfi-7) X f 7+ 12". 



Velocity of Rope in Feet per Minute. 































Table XXII 

. — Rope Pulleys for 

General Work. 

Diameter of Rope. . 







Diameter of Pulley 








It is well to remember that the cause of many failures 
and mucji trouble experienced in rope-driving is due to 
the use of too small a pulley for the size of rope and ten- 
sion carried, but we have yet to hear of a case where the 
diameter of pulley has been too great. AVhen it is not ab- 
solutely necessary to restrict the diameter to the smallest 
possible which may be used, the least diameter should con- 
form to that given in Table XXII, which has been arranged 
for general work, and gives least diameters, to the nearest 
half foot, for rope-pulleys suitable for all speeds within the 
limits of good practice. 

When a close velocity ratio between driver and follower, 
greater or less than unity, is required, the pitch diameter 
of each pulley should be measured from the point of tan- 
gency of the rope in the groove, and not from the centre 
of the rope. For, if D represent the diameter of driver 
measured to centre of rope (Fig. 63), and F the corre- 
sponding diameter of follower, the velocity ratio will be 

£) 2x 

-^ — ^r-5 where x is the common vertical distance between 
F — 2x' 

centre of rope and point of contact. Where the pulleys 

are the same size it is obvious that the velocity ratio will 

not be changed, but as the difference in diameter increases 

the influence of x will be more marked; thus it D = SF, 

the velocity ratio will be -^ — — , — a result manifestly 

greater than -^. In any case, the smaller the diameter of 



pallejs for a giTen Telocity ratio, the greater the effect of 
the qaantitj x. 

In ordinary single transmissions it will be sufficiently 
close to assame the diameter of palley as measured from 
centre of rope. 



Fig. 62. — Diameter of Pulley. 

It is evident irom the previous considerations that the 
condition and shape of the groove is a matter of much 
importance: so generally is this recognized, that manufac- 
turers now almost universally turn their rims to special 
gauges and templets, which insure uniformity in diameter 
and shape of groove. Special tools have been devised for 
turning the grooves, and one prominent manufacturer has 
constructed a special machine in which the rims are milled 
out. By this process some of the smaller pulleys are 
machined in one operation — as many cutters being used as 
there are grooves to be milled. Uniformity of pitch, diam- 
eter, and contour are thus insured independently of the 
operator. Formerly the bottom of the groove was fur- 


nished with spikes, or the sides were cut into angular teeth 
in order to prevent the rope from slipping;* but a greater 
experience with rope-driving has shown that the whole sur- 
face of the groove must be perfectly smooth, and should 
be carefully polished as well as machined, since the fibres 
of the rope, if allowed to rub on a rough-turned or cast 
surface, will gradually break, fibre by fibre, and thus give 
the rope a short life. It is also necessary to avoid using 
any pulley with sand or blow holes in the groove, as they 
are very destructive to a rope: when blow-holes occur, if 
not honeycombed excessively, they should be filled in with 
lead, or, preferably. Babbitt's metal; otherwise the pulley 
should not be used. 

Some rope-pulleys are simply cast and the rims smoothed 
up by holding a piece of abrasive material, such as a bro- 
ken emery-wheel or grindstone, against the surface of the 
groove while the pulley revolves at a high speed. While 
this may produce a smooth surface, such an expedient is 
doubtful economy; for no matter how carefully a multiple- 
grooved pulley may be cast, it is almost impossible to ob- 
tain a rim all the grooves of which are true and of the same 
diameter: the result is that, with such pulleys, the ropes 
tend to vibrate and sway from side to side, rubbing against 
each other, and frequently against side-posts, walls, floors, 
and other obstructions, which rapidly destroy the rope. 

Attempts have been made to produce a finished surface 
in the groove by casting the rim in an accurately turned 
chill, but such pulleys have not been as successful as an- 
ticipated. A form of pulley made by the Link-Belt Engi- 
neering Co. of Philadelphia consists of an iron sheave cast 
in rings, some with and some without arms and hub, from 
which a complete pulley is built up having the requisite 
number of arms for strength, and at the same time being 

* Willis : *' Principles of Mechanism." 



light and free from excessive shrinkage strains liable to ex- 
ist in wheels having light arms and heavy rims and hubs. 
The aheaves have a alight projection 
on one side and corresponding re- 
cess on the other, with bolt-holes at 
the circumference, ao that a niiiltiple- 
grooved pnlley with any number of 
giOovcB may be readily built up by 
' bolting the sections together as shown 


Fio 6a — Bl II t I. p Rope Pulley for Light Work. 
inFig.63. Thtse wheels aie niiide from metal patterns and 
moulded in a three-part iron flask. Tlie core which makes 
the groove is continuous and of green sand, prodncing a 
very smooth casting. The groove is finished with an em- 
ery-wheel swinging in a frame like a cut off saw, the final 
finish being given by the use of emery and oil. While this 
process produces a single-grooved wheel more cheaply than 
by turning, it is doubtful whether from a manufacturing 
standpoint there is any economy in the built-up wheel 
when we consider the degree of accunicy now obtained in 
moulding multiple-grooved wheels, and the consequent re- 
duction of labor in turning them. 

Another speciiil form of ropc-driving pulley is that made 
by John Musgrave & Sons, Bolton, England. This pulley 
(Fig. 64), it will be noticed, is extremely light, but is sufB- 
ciently strong for the requirements. This lightness is at- 
tained by the use of steel arms turned tapering, and firmly 
secured to the rim and the hub, which latter is split in 


three segmenta and ringed with steel. By the use of this 
form of pulley the shafts are relieved to h coDsiderahle 
extent of the weight urid consequeDt* friction entailed by 
the ordinary pulley, and there are no escessive shnnkage 
strains to contend with, as is usuiilly the case with the 

Fig. 64. — Rope Pvumy 

common form of single-casting jnitley. In pulleys of tliis 
form, where wiought-ii'on and stfel rods are used for arms, 
the ends of the rods should be dipped in acid and tinned 
before setting in the mould. In addition to this the rods 
are frequently headed up and grooves turned in the ends. 
The bottom of the groove in casl. iron pulleys is gomt;- 


times filled with wooden blocks dovetailed into a channel 
cast to receive them, in which case the rope runs on the 
bottom and the shape of the groove approximates that 
shown in Fig. 65; after 
being fitted and secured 
the groove is trued up and 
turned out to the desired 
jhape. Gutta-percha, rub- 
/ ber, leather, tarred hemp, 
and other materials have 
been used for the same pur- 
pose; bnt in the best mod- 
ern practice a smooth cast- 
iron surface ie pi-eferred to 
any other, and we find these 
groove linings confined chiefiy to pulleys used in wire-rope 
transmissions and hoisting-machinery. 

Of late years an all-wood ritn with V or U grooves has 
been used to some extent, as it makes a cheap pulley, and 
is very satisfactory for light work when a semicircnlar 
groove is adopted. Wooden rims, Iiowever, should not be 
used for heavy work, and the slip of the rope should be re- 
duced to a minimam ; otherwiee the grooves and ropes will 
be rapidly cut out and the whole system will he very unsat- 

The proportions of rope-pnlley rims depend somewhat 
upon the shape of groove adopted. Some manufacturei-s 
have a standard groove, with straight sides, which may be 
used for several different-sized ropes, as shown in Fig. 6C. 
For light work snch a pulley is very satisfactory, and has 
many advant-nges in regard to constructive features not 
possessed by any other form. The groove shown in Fig. 
67 * is very commonly used in England, but in this country 

* Low and Bcvis, " Mnrbiuc Design," p. ISfl 


manufacturers of large rope-pulleys usually prefer a form 
in which the abrupt change in profile (:ia at n) does not 
occur. Of these the more common form is a modificatiou 

Pio 86 —Rim SEcrroHS— STUAicnT Groove 
of tht. inglisli section, in which the straight sideb forming 
the angular groove are connected to the rib between the 
grooves by cuives as in Fig 68 

Anothei form which has much in its fivor especially 
for the independent rope system, is the circular arc groove, 
in which the sides are formed by area of circles. 


Working proportions for this groove are given in Fig. 
19, in wfaich the unit d equals the diameter of rope. It 

Pto. 68.— Rim Section— iMpnovED English Fork, 
will be noticed tliat tlie centre for the curve is located at 
the inters(!ctio]i of a line drawn through the centre of the 



rope at an angle of 22^° with the horizontal ;ind a line 
druwn throngli the tops of the dividing tibs; the angle of 
the groove embra':ed by the rope ia thus dopendont upon 
the position of the latter in the groove: in its normal posi- 
tion the angle is one of 45°. Thu Walker groove, as now 
used in lope-pulleya made by Fraser & Clialmers of Chicago 
and the Walker Mfg. Co. of Cleveland, has its sides formed 
with circular arcs similar to Fig. 69, but the angle of the 

groove is more acute; at the point of tangency when the 
rope simply ^-esta in the groove the angle measures 33°. 

Kope-pulleys for ordinary transmissions ar? nanally flat 
on the inside of the rim, or slightly tapered as shown by 
the full linea in Fig. 69. In large wheels, however, the 
rim is frequently swept up, and where the weight with a 
flat rim would be greater than required for strength or 
steady running, the inside is hollowed out as shown by 
dotted line. This givea a more nearly uniform section 
and makes a stronger and lighter wheel, but the expense of 
construction is greater. Guide-pulleys or idlers should be 
made with a semicircular groove, so that the rope rnns 
upon the bottom instead of being wedged between the 
sides, as ia grip-pulleys. Some engineers maintain that 



the wedge groove sliould be used in all cases, and many 
plants are in operation having V-shaped grooves in the 
idlers; but we believe the most satisfactory results will be 
obtained by using a groove in which the rope runs on the 
bottom. Of these there are two general forms — the one 
in which the rope has considerable play in the groove, and 
the other of such form that the rope is embraced by a 
portion of the surface of the groove: the first is used more 
particularly for single ropes, while the latter is adapted to 
any number of wiaps. 

A modification of the first form is frequently used for a 
number of rojies as shown in Fig. 70. 

Fig. 70.— Form op Groove for Guide Pulleys. 

The proportions shown in Fig. 71 will be found very 
satisfactory for single or multiple groove guide-pulleys, in 
which the pitch is equal to that used with the grip-pulleys 
represented in Fig. 69; the general design conforms closely 
to the latter, bnt the pulley is somewhat lighter. 

For shaft-pulleys and the smaller sizes of fly-wheels not 
exceeding about nine feet in diameter the casting is usually 
made in one piece, unless a '^splif pulley is required. In 
order to relieve the wheel in cooling the arms were formerly 



given a curved or S shape, as this form yields more readily 
and is supposed to conform to the iinequul contraction of 
the rim and hub of the pulley in cooling. With properly 
proportioned wheels, however, and with due care in the 
foundry, striiighU armed pulleys may be cast as strong as 
those of curved ontliae, and as the former are lighter, neater, 
and cheaper than the latter they are now almost universally 
employed. In the larger single-casting pulleys the hubs 
are frequently split in order to favor the arms m cooling: 

Fro, 71,— Form of Groove for Guide Puixets. 

when split by a diametral plane the two portions at the 
hub are generally secured by bolts and nuts as in Pig, 73, 
or less frequently by pins and cotters; when split in three, 
wrougbt-iron or steel rings are shrunk onto the ends of the 
hub, which is turned to receive them, as represented in Fig, 
73, Occasionally rings are shrunk on and bolts used as 
well, but this precaution is not common practice. 

There is no general rule by which the number of arms 
maybe determined. For small rope-pulleys the number is 
nsually six, while for the larger sizes six or eight arms are 
naed for pulleys cast in one or two parts; but these, in some 



cases, hare two sets of arms as shown in Fig. 74, which 
represents a 78-inch pnllej made hy the Kohert Poole & 
Son Co. for the Proyidence Cahle Tramway Co. The nsnal 


Fig. 72. 

Fig. 73. 

run of wheels from 9 feet to 15 feet in diameter are cast 
in halves, six or eight arms being employed. 

Occasionally, however, narrow rope fly-wheels, of diam- 
eters up to 18 feet, are made in two pieces only, with eight 


Moderate-sized wheels arc frequently cast in one, bnt 
arranged so as to be readily split in two afterwards. 

Pulleys cast in halves having light rims should have 
double arms along the line of separation, as shown In fig. 
75, which represents a form of split pulley made by Fair- 



banks, Morse & Co.; this prevents undue bending action, 
and is otherwise the better able to resist the effect of cen- 
trifugal force due to the added weiglit at the point of 

For diameters ranging from 15 feet to 20 feet the usual 
practice is to make the pulley in six or eight segments with 
as many arms, although some makers prefer ten segments; 
from 20 to 26 feet ten segments are usually adopted. For 
wheels from about 26 to 32 feet twelve segments and as 
many arms are generally used, while larger-sized wheels are 
built up of fourteen, sixteen, and even more segments — 
sometimes as many as twenty-four being employed.* 

When rope-pulleys are made in halves or are built up it 
is important that the connecting bolts and flanges in the 
rim should be strong enough to resist the maximum stresses 
that may occur in the joint. 

Ordinarily in rope-pulleys the net section of bolt area is 
about 12 per cent of the section at the joint, but this ranges 
from 25 per cent to about 6 per cent in different cases. 
Various shop rules are used by designers for obtaining the 
bolt section, many of which give the bolt area directly in 
terms of the cross-section of the rim. While such formulae 
may give satisfactory results for an assumed average rim 
speed, it is better to design the joint in large wheels from 
a consideration of the particular conditions in each case. 

In very slow-moving heavy wheels the principal stress 
may be that due to the weight of the wheel itself, but in 
rope-pulleys and fly-wheels the rim has usually a high 
velocity, and the strain due to centrifugal force in the rim 
is the principal factor in determining the bolt section ; it is 
obvious, however, that in high-speed heavy wlieels both 
influences should be taken into consideration. 

The tension in the rim produced by centrifugal force is 

* American Machinist, Feb. 16, 1893. 


equal to one half the centrifugal forc« dnc to tUe weight 
iiiid velocity of the rim tnaltiplied by the ratio of diameter 
to semi-circamference, or eqnal to 

As tliis force ia resisted at each end of a diameter, the 
strain TV, or hoop-tensiuii, acting at either end will be one 
half the above: hence 

in which F, has the usnal value 0.00034 WRN*, 
where W ^= entire weight of rim; 

R = effective radius of rim in feet; 
jV = revolutions of wheel per minute. 
The tension at each end of a diameter due to the weight 

of the rim is evidently equal to sf-^-j ; therefore the stress 

in the rim due to F^ and ITwiU be 

If the bolts could be placed at the point 
of application of this force, their eSective 

section A would be -, where/is the allow- 
able stress in the bolts. 

In the actual construction the centre of 
holts must be placed at some distance 
from the point of application, in which 
case the bolts arc snbjucted to an addi- 
tional strain due to the bending moment 
at the joint, us shown in Fig. 76. If this bending 
nionieut is taken up by the rim, the bolt section will be 

Pio 76. 


j2i& before; but if the rim and bolt flanges are rigid and 

resist any deformation, the total strain comes upon 
the bolts. Under these conditions the net section A 

may be obtained by multiplying j by the ratio of the 

leverage y of the force 8 to the leverage x of the resist-J|^^ 

ance in the bolt; hence if we assume that x = ^y, a com- 

mon value, we shall obfcain ^ = 2^. 

By using studs with nuts at each end the bolt may bo 
placed nearer the rim, in which case, for the same depth of 

flange, the ratio - may be made less than 2. In order to 


obtain ample strength at the rim, the bolt section should 
be determined on the supposition that the full bending 
moment will be thrown upon the bolts. In this case the 
bolt-flanges or lugs must be able to resist any bending due 
to the force S, The bolt-flanges will be as strong as the 
rim if we make the respective moments of resistance equal 
to each other; hence 

\ih = face of wheel; 

V = breadth of bolt -flange minus the width of bolt- 

t =■ mean thickness of rim; 

i' — thickness of bolt-flange or lug, 

ibty=ib'iy or i' = i/^,. 

This thickness t' may be reduced somewhat if strength- 
ening ribs are carried from the rim to the lower edge of 
the bolt-flange, as shown in Fig. 76. In the above, t 
was taken as the mean thickness of rim reduced to a rec- 
tangle, and no account has been taken of the strength- 
ening ribs and flanges. If the leverage of the bolts is 

assnmed as one h^'f. we mar obLiin the total net section 
of the bolts as fol>jvs: 

Since P, = 0.0<X»:J4 FT^.V, 

^ = i| O.OXHOS WR 


In the aboTe formula / may be taken equal to 6000 
ponnds, for lx>Itd np to 1^ or 1^ inches diameter^ bnt for 
larger sizes 8000 to 9000 ponnds may be nsed. 

If ^ = the mean thickness and h = breadth of rim in 
inches, the weight W may be determined from 

W=2nRy. 12/6x0.26 
= 19.6^/6. 

If the rim is properly bolted, the principal strains to 
which the hub-bolts are subjected are those due to the 
weight of hub and arms and the tension produced by key- 
ing to the shaft. 

As the weight of hub and arms in large pulleys is usu- 
ally much less than the weight of rim, it will be seen that 
the strain on the hub-bolts due to the weight and centrif- 
ugal force of these parts will be less than that on the rim- 

In practice some makers design the hub-joint so that the 
net sfiction of all the bolts is equal to the bolt section in 
one edge of the rim-joint; this, however, is not usual 
l)ractice. An inspection of a great many pulleys made by 
various manufacturers shows that the hub-bolt section is 
often twice as great as the rim-bolt section; but in many 
of those cases the rim-bolting is very weak. 

It is safe to make the bolt area the same in each case ; 
l)\it if the rini-joiiit is made according to the method 


indicated, tlie bolts in the hub vill usually be safficiently 
strong if their totitl effective section is equal to that in 
one rim joint: for light pulleja, however, this should be 

The section of arm for tliose pullejs in which the arms 
and hub are cast together is usually elliptical or segmental, 
as shown in Fig. 77, 

Fig. 77.— Sections of Arms. 

The elliptical form as here given is so proportioned that 
Its minor axis is one half its major; in the segmental form 
the minor axis is four tenths the major. 

The crosB-section of the arm may be determined by con- 
sidering it as a beam fixed at one end and loaded at the 
other with the force P due to the pull of the ropes. The 
bending moment on each arm will then be 

asEuming that the load is divided equally among the num- 
ber of arms, JV„. 

• Fur complete analysis of strains in fly-wheel rims, see Prof, tJn- 
win's discussioD in liin "Macliiiie DeM^n," vol, ii. Also. Mr. 
Slanwood'a paper jd vol. xiv. Tmns. A S. M. E.. with Prof. 
Lanza's diecussion. 


Since the bending moment is equal to the modulus of 
the section^ Z^ multiplied by the stress,/, in the material, 
we have 

For an elliptical section whose major axis is h and minor 
axis 0.4A, 


f =/X3^X.4A'.. .... (18) 

If we let 7,, the maximum tension in the rope, equal 


200^" pounds, and -j^ = 2, we shall have the load acting on 

each arm 

P = 100rf"/t, 

where n is the number and d the diameter of ropes. 
Therefore by substituting this value of P in equation (18) 

there is obtained h = 1/]^^^^ 

Assuming that/ = 2500 pounds per square inch for cast 
iron and that N^ equals 6, we have 

h = 0.44 \/¥nD, 

where D = pitch diameter of pulley and h is the major 
axis of arm produced to centre of wheel, both in inches. As 
the smaller pulleys require a larger margin of strength, on 
account of their greater liability to breakage in handling 
and casting, the above formula should be modified by intro- 
ducing a constant; if we let this constant equal \ inch, a 
suitable value for the width of arm will be obtained from 

A = 0.44VS^wF+0.5" (19) 


As the centrifugal force set up in the rim of the pulley 
causes an additioual stress in the metal, the value of / 
should be chosen with reference to the speed at which the 
pulley is to run. The higher speed not only induces a 
greater stress in the material, but the liability of failure 
due to vibration or shocks is greatly increased. The stress 
in the rim due to centrifugal fofce may be obtained by 
considering the pulley as a cylinder subjected to a force 
p[ = F^) per square inch. The thickness of a thin cylinder 
to resist rupture may be obtained from 

_ pD'' ___ pr 

in which t = thickness in inches; 

jt? = pressure per square inch; 
r = radius of cylinder in inches; 
f = allowable stress in pounds. 

In this case 

W= weight in pounds = 0.261 pounds per cubic inch; 
V = velocity of rim in feet per second; 
g = 32.16; 

R = radius of rim in feet = — ; 


^_ 12W 



If F^ act on one square inch of p alley whose thickness} 
18 unity, we shall hare 

^ 0.261 y \1i^ ^r^»^ 
f = .T^^. = 0.097v" 

32. IG 


= Y^, very nearly; 

this is the stress in rim per sqnare inch of section due 
to centrifugal force alone. 

When V in feet per second = 
'• r** " ** minute = 

8treg8 f in pounds due to centrif- 
ugal tension = 






70 80 
4-.iO0 4800 

490 640 











The stresses due to the pull of the ropes, and those dne 
to contraction in cooling, are additional to those here given, 
hence / should be taken sufficiently low to allow for the 
various stresses which may be set up in the pulley. If we 
assume that the working stress should not ordinarily exceed 
3500 pounds per square inch for cast-iron pulley arms, and 
that it should be less for high speeds where the dynamic 
effect of shock and vibration is greater, a suitable value 
for cast iron may be obtained from the empirical formula 

/ = 


5+ fT 

From this formula the annexed values of / have been 
calculated; it will be noticed that the value of/ used in 
formula (19), namely, 2500 pounds per square inch, corre- 
sponds to a velocity of rim equal to about 3500 feet per 

VoU>oltv of i\>pe iu feet per minute r= 
Allowable HtretS8 / = 








The arms should taper toward the rim -^^ inch per inch 
of length, that is, ^^ on euch side, but in no case should 
ihe width of arm at the rim be less than two thirds its 
width at the centre of shaft. 

Very wide pulleys in which the proportions for single 
arms would be inconveniently large may be made with two 
or three sets of arms; in such cases they may be considered 
as two or three separate pulleys combined in one, except 
that the proportions of the arms should be 0.8 to 0.7 times 
that of single arm pulleys.* 

In designing the hubs of wheels practice varies consider- 
ably. Some authorities give the thickness of metal around 
the eye in terms of the pulley diameter only, others take 
into account the diameter of pulley and also the diameter 
of shaft or the breadth of face, while the length is variously 
given in terms of the diameter of shaft or the face of 
pulley, or both. 

For rope-pulleys if the thickness of metal in the hub is 
made proportional to the diameter of pulley and also to the 
diameter of shaft, and if the length of hub is made to vary 
with the number and size of ropes and the diameter of 
shaft, we believe the requirements of strength and good 
proportions will be best attained. 
If 2) = diameter of pulley, 
d, = " " shaft, 
d^ = " " hub, 
d = " " rope, 
n = number of ropes, 
Lf^ = length of hub, 
then the diameter and length of hub may be obtained 
from the following formulae, which have been deduced from 
the proportions of a large number of rope pulleys made by 
representative manufacturers: 

d^= 0.026 D + ds + 2i'' 

* Reuleaux, ** Constructeur." 




Lf, = 0.6 J/i + d, + i". 

For loose or idle pulleys the diameter of hub may be 
made less than that given by the above formula, which 
allows for keying. In general the thickness of metal 
around the eye in an idle pulley may be taken as about 
two thirds as great as that in fixed pulleys of the same 
diameter and face. The length of hub in idlers should be 
sufficient to give a good bearing surfjice, and may vary 
from two to three times the diameter of shaft— depending 
somewhat upon the speed of rotation. 

Fig. 78.— Method op Joining Abms and Rim. 

In the construction of large rope-pulleys which are 
made in segments, the usual method is to bolt the rim sec- 
tion to the arms at the ends of the segment, as shown in 
Fig. 78. 

When the rim segments are joined midway between the 
arms as in Fig. 79 the several connections are simplified to 
some extent, but with this method of connection there is a 
decided tendency for the joint to open under the influence 
of centrifugal force, which is increased somewhat by the 
weight of the connecting flanges and bolts. With the rim- 
joints at the junction with the arms, however, a rigid con- 



nection between adjaoent arms is obtained, and the centrif- 
ugal effect of the rim tends to increase the tension in the 
arm without opening the joint. 

Examples of both methods of connection are given in 
Figs. 80 to 93, which also represent some of the details of 
construction in various built-up rope-pulleys. 

As will be noticed, these built-up wheels are made with 
a large central boss or hub, usually in one piece, but some- 
times in two, provided with seats for the arms, or bored. 

Fig. 79. — Method of Joining Arms and Rim. 

either tapering or straight, to receive the ends of the arms, 
which are turned to fit the holes. 

The arms are usually of round or elliptical section, cast 
hollow; but other forms are common. Thus in Figs. 85 
and 90 the arm is of cruciform section, while in Fig. 81 a 
modification of the H section is used. 

In Fig. 80 we have an interesting form of wheel patented 
some years ago by Mr. James Barbour, of Combe, Barbour 
& Combe, Belfast.* The rim of the wheel is made in seg- 

* Engineering, Sept. 7, 1888. 



ments^ and is attached to the boss or hub by means of the 
bolts or tie-rods dy extending from the rim to the boss as 
shown, and passing through the arras, which are made hol- 
low for the purpose. The rim and hub are recessed to re- 
ceive projecting pieces a'a' on the ends of the arms. The 

Fig. 80. — ^Built-up Wheel. (Barbour.) 

head of the tie-rod d' can be made to form part of the 
periphery of the wheel. The tie-rods have slots formed in 
their inner ends to receive keys h\ which keys are passed 
into the hub through holes parallel to the axis of the shaft, 
din this way the rim of the wheel is secured to the hub. 


SFt > 


The object of the arrangement is to obviate the use of 
a number of bolts for connecting the rim to the arms; 
moreover, the tie-rods withstand the centrifugal force of 
the rim, and the wheel can be driven \iith safety at a 
high rate of speed. The usual projections from the 
principal surfaces are also eliminated, and the currents 
of air generated by the high velocity of the rim are less- 
ened. This latter feature is one the value of which is be- 
coming more recognized lately, especially in rope-driving, 
when a high velocity of rim is especially advantageous. 

To lessen the fan action set up in a large fly-wheel, we 
frequently find the arms boarded up, provision being made 
for this in the wheel. 

Fig. 81 represents a fly-wheel 30 feet in diameter made 
by Hick, Hargreaves & Co., Bolton, for a 1000 horse-power 
condensing-engine.* The weight of the wheel is 54 tons. 
The rim is 6 feet wide, and is grooved for 2 7 cotton ropes 
5 inches in circumference, or nearly If inches in diameter, 
at 2^ inches pitch. The grooves are shown in detail 
at (a). 

The rim is constructed of twelve segments with twelve 
arms. The segments are planed at the joints, and are 
bolted together with eight bolts and nuts, of which the 
two next the arms are If inches in diameter and the others 
are 1 J inches. Each arm is secured with four 2-inch T- 
head bolts and nuts to the rim segments, two to each of 
two segments. The boss, nave, or centre, shown in sec- 
tion at (b), is 6 feet J inch in diameter, or 7 feet across the 
platforms, or slightly raised plane surfaces, on which the 
arras take their bearings. It is 18 inches wide at the rim 
and 2 feet wide at the bearing on the shaft. Twelve sock- 
ets are bored out to receive the ends of the arms. The 
arms are formed approximately of H section, as at (c), and 

D. K. Clarke, ••Steain-Engino," vol. lu. 


measure 14 inclies by 9 inches at tlie centre and 9^ incliee 
by 6f inches at the rim. They are turned conically to fit 

Pig, 82. — Twbnty.f< 

r RoPK-w 


the holes in the centre, tapering from 12 inches in diam- 
eter at the outer ends of the soclteta to 7^ inches at the 


inner ends in a total length of 2 feet 5 inches. The arm 
is keyed into the centre with two cotters, each 24 inches 
long, 1 inch thick, tapering in width from 3f inches to 3i 
inches, or i inch in 24 inches. They are driven in one 
from each side of the nave reversely, and make np a 
united width of 6| inches. To join the rim the arm is ex- 
panded into a flat flange 21 inches by 18 inches and 4 
inches thick, through which the four bolts already named 
are passed. The opening at the centre is 25 inches in di- 
ameter, or 2 inches larger than the shaft. The wheel is 
fixed on the shaft with six keys 5 inches wide and If 
inches thick, bearing on six flat seats formed on the shaft, 
with a taper of i inch to the foot. 

Fig. 82 represents a somewhat similar method of con- 
nection employed by the Walker Manufacturing Company, 
Cleveland. In this pulley the arms are of round or pipe 
section, and present a much neater appearance and better 
proportions than obtain in the previous pulley. 

This wheel was made for the Baltimore City Passenger 
Railway Station, and is 24 feet in diameter, 30 inches face, 
grooved for ten 2-inch ropes at 2f inches pitch. The de- 
tails are shown in Figs. 83 and 84. 

The centre is a single casting 5 feet 6 inches in diam- 
eter, bored to receive the arms, of which there are ten. 
Each arm is 10^ inches in diameter near the hub and 8 
inches in diameter near the rim, where it is expanded into 
a flange and bolted to the rim segments. The bolts con- 
necting the arms to the segments and the segments to 
each other are all turned 1 j| inches in diameter, and fit in 
drilled and reamed holes. 

The shoulder on the arms at the hub is 15 inches in di- 
ameter, and is faced to fit the machined seats on the boss. 
By this means perfect alignment is obtained, and no 
motion of the parts is possible unless the key shears. 

Fig. 85 represents a rope-pulley designed by Mr. F. Van 


Fio. 88.— Details or Put.let bbovn nr Fig. 83. Section op Arm. 


I. 84.— Detao-h of Pri,LEy shown in Fto. 83. 




Vleck for the San Di'rzo Cable-railroiui Pover-siarion.* It 
U 25 fc-t ■!! diame'.er. -I'ij ;n>;ae« face, grijoverl for twelre 
2-incli CitUtD ro|--5 at 3j iocbes pitch. This dmni is com- 
posed of t^n separate segmenia and ten urns, botwd uid 
keyed midway in each segment. The central haih is 4^ 

feet in diameter across flats, and is proportioned, as in fact 
the whole wheel is, with epecial regard to lightness. 

The designs shown in Figs. 8C to 89 are for the large 
rope-pulleys used in the Fifty-first Street and Houston Street 
" TranB. A. B. M. E., vol. xii. p. 77, 

.Thirty-two-foot Rope wheel. (Walker Mfg. Co.) 


Fig. 88.— Thihty-two-foot Rope-whebl. (Walker Mfg. Co ) 



stations of the Bioadway Gable Road. The first wheel, 
Fig. 86, is 32 feet in diameter, 37^ inch face, and is grooved 
for 13 cotton (Lambeth) ropes 2 incbes in diameter, 3f 
inches pitcli. The speed of the ropes is very low — only 
1877 feet per minute. Mr. M. W. Sewiill* gives the fol- 
lowing pai'ticulars regarding the details of thie wheel, as 
at first designed. It will be seen by reference co the flgiire 
that the arms of the drnm are secured to the centre by 
clamping two tnrned portions on the inner end of each 
arm, between heavy cast-iron disks, one of which is cast 

as a fiange on the hnb and the other acta as a fol- 
lower. These are bolted up a slight distance apart, and 
the holes for the arms are bored accurately to fit the turned 
portions of the same. The bolts are then loosened, the 
arms put in place, and the follower bolted hard up to the 
hub fiange. A taper pin is then driven into a reamed hole 
through each arm and the parts of the centre, and held in 
place by a nut. The arms and tlie centre are then practi- 

* For complete description of the cable- drivtng macliiBery Bee 
dmeriean Machinist, t/lny %iua>\ 31, lti»4. 



arm is accurately fitted. In addition to this two of the four 
bolts securing each arm to the centre is turned to fit reamed 
holes in the boss ; these bolts thus act as dowels and pre- 
vent any working of the parts. The bolts are calculated 
to resist the maximum tension in the arms, and are 2^ and 
2^ inches in diameter ; in the same way the arms are bolted 
to the rim, two of the four bolts fitting in reamed holes. 
The arms are of hollow elliptical section 15 by 10 inches at 
the hub, and 10 X 7i inches at the rim. Each centre is 8 
feet diameter across flats ; thev are bored 20 inches in di- 
ameter, and are 3 feet long in the bore. 

Large rope-pulleys from 20 to 32 feet in diameter are, 
when extra wide, frequently made with two centres as well 
as two sets of arms, as just noted in Fig. 87. This is also 
seen in Figs. 90 to 92, which represent a rope-driving wheel 
26 feet 9 inches pitch diameter, 8 feet face, grooved for 
twenty-four 3-inch ropes, at 3J inches pitch. This pulley 
was designed and built by Robt. Wetherill & Co. of Ches- 
ter, Pa., and consists essentially of two separate and inde- 
pendent drums, flanged and bolted together at the rim. 
Each centre is made of two separate disks 6 feet 6 inches 
in diameter, bored and faced on the inside. The arms, of 
which there are twelve, are of cruciform section between 
the boss and the rim, where they are flanged and bolted to 
the rim segments in the usual manner. At the centre the 
arms are wedge-shaped, 8 inches thick, and are so propor- 
tioned that when accurately planed and fitted they form a 
complete circle. 

These arm segments are then bolted between the two 
centre disks, and make a strong and compact hub. 


Actaal section of ropes, 92 

Adhesion of ropes, 168 

Advantage of high rotative speeds, 157 

Advantages of rope-driving, 4, 5 

American system of rope-driving, 24, 25 

Angle embraced by rope, 117 

Angle of groove. 164 

Area of ropes, 92 

Arms, number of, in pulleys, 190 

Arms, shape of, 199 

Arms, taper of, 203 

Atlas Mills, 12 

Atmospheric changes, 81 

Automatic tension- carriage, 28 

Axial rotation of ropes, 174 

Barbour, James, built-up pulley, 206 

Barrus, G. H., friction in mills, 7 

Beeswax on ropes, 89 

Belts, leather, 1, 34, 40 

Blow-holes in pulleys, 183 

Bolt area for pulleys, 195 

Braided rope-joint, 23 

Broadway Cable Road, built-up pulleys, 217 

Brown, A. G., friction loss, 6 

Brush Electric Light and Power Co., Niagara Falls, 56 

Built-up pulleys, 183, 195, 205 to 221 

Catenary curve, 8, 12 
Cast grooves, 183 
Centrifugal force, 45 

Centrifugal force, influence of. 111, 119, 163 
Combe & Barbour, early rope-drives, 2 

Combe, Barbour & Combe, basis for calculating horse-power, 99 


224 INDEX. 

('otton ropes, 77, 81 

Cotton fibre, 78 

Cotton wax, 78 

Continuous-wind system, 25 

Cone-pulleys for ropes, 35 

Coil- friction, 44, 48 

Corliss engine and u&e of jack-shaft, 41 

Coefficient of friction, 112 

Cost of ropes, 101, 122 

Coulter, Dr. S. M., tests on manilla fibre, 83 

Coupling, cut-off, 27 

Coupling for braided rope, 28 

Creep of ropes, 174 

Cross-section of rope, 92 

Cut-off coupling, 27 

Deflection of rope, 131, 170 
Degree of twist in ropes, 86 
Details of rope-pulleys, 205 to 221 
Diameter of bolts for pulleys, 195 
Diameter of pulleys, 103, 161, 177 
Differential driving, 36, 170 
Differential pulley, Walker's, 173 
Doable arms in pulleys, 193 
Double ropes recommended, 108 
Draw-rods, 71 

Durie, James, on rope-driving, 8 
Dyblie's rope-tightener, 30 
Dynamo- driving, 38 

Early use of ropes for driving, 2, 64 
Effect of centrifugal force, 45 
Effect of tar on ropes, 91 
Effect of wedging in groove, 162 
Efficiency in any given case, 73 
Elasticity of ropes, 4, 162 
Elastic slip of ropes, 174 
Engines, friction in, 6, 142 
English rope system, 4 

"oe transmissions, 96 
•a friction, 141 
ige rope- wheels, 205 to 221 

IKDEX. 225 

Fairbanks, Morse & Co., split pulley, 198 

Fibrous ropes, 75 

Fibre, cotton, 78 

Fibres of manilla, 83 

Flax ropes, 77 

Fly-wheels, heavy, 5 

Frictional grip, 167 

Friction and stress moduli, 118 

Friction-clutch, 83, 43 

Friction, coil, 44, 48 

Friction, coefficient of, 113 

Friction loss, 141, 158 

Friction of engines, 6 

Friction of shafting, 6, 6S. 145 

Gear wheels, 1, 5, 8 
Graphite on ropes, 89, 91 
Gregg, multiple sheaves, IG, 38 
Groove, angle, 104 
Groove, shape of, 188 
Groove, surface of, 183 
Groove, wedging of rope in, 163 
Guide-pulleys, 189 

Harmonic vibration, 105 

Heavy fly-wheels, 5 

Hemp ropes, 3. 77, 90 

Henthorn, J. F., friction in mills, 7 

Hick, Hargreaves & Co., built-up pulley, 208 

Him, C. F., transmission of power, 63 

Hoadley Bros., Power-house^ Chicago City Ry. Co., 53 

Horse-power of ropes, 111 to 131 

Hubs of pulleys, 191, 308 

Hunt, C. W., form of splice, 30 

Idlers, 34, 86 

Inclined transmissions, 189 

Influence of belt pull, 157 

Influence of centrifugal force. 111, 119, 163 

Introduction of rope-driving, 3 

Jack-shaft, use of, for dynamo-drives, 38, 43 

226 INDEX. 

J*w cXaieh, 42 

Joint for braided ropes, 23 

Kircftldy, te«t« on ropes, 80 

I^ambetb ropes, 80, 88 

liftneti Mills, 12 

Large rope-wheels, 205 to 221 

Ijaxey, large overshot wheel at, 71 

l^ast diameter of pulleys, 103, 161, 179 

Leather belts, 1 

I^ength of roanilla fibre, 86 

Life of ropes, 107 

Limit of length in shafting, 69 

Linseed-oil on ropes, 89 

Link-belt Co., Western Electric Co.'s PUint, 25 

Link-belt Co., Virginia Hotel Plant, 42 

Liverpool Overhead Railway, 40 

Lockwood & Greene, Lanett Mills, 12 

l>jckwood & Greene, Naumkeag Mills, 12 

Ijong-distance transmiHsion, 63 

Loss due to Ijending. 159 

Loss due to friction, 141, 158 

Txjss due to winder-pulleys, 50 

liossesin ropes, 141, 159 

Lubrication of ropes, 79, 88 

Manilla fibre, 82.86 
Manilla ropes, 77 

Marlin-Hpike, improved form of, 23 
Methoa of joining arms and rim, 204 
Miller, 1'. S., plant of Western Electric Co., 25 
*t *t varying angle of groove, 168 

" *' use of small ropes, 100 

•* " use of loose idler, 36 

Multiple idle sheaves, 80, 88 
Multiple-rope system, 4, 17 
Musgrave & Sons, Atlas Mills, 12 
<• ** •• , Nevsky Mills, 16 
" *• •• , rope-pulley, 185 
M •« " , life of cotton ropes, 107 

INDEX. 227 

Naumkeag Mills, 12 

Nevsky Mills, 16 

Normal working load, 98, 100 

Outdoor transmission, 50, 66 
Overshot wheel at Laxey, 71 

Pine tar as lubricant, 89 

Pitch diameter of pulley, 181 

Power absorbed by friction in shafting, 158 

Power absorbed by ropes and gears, (i 

Power transmitted by shafting, 15$ 

Pulleys, diameter of, 38, 103, 161, 177. 181 

Pulley, tightener, 48 

Pulleys, supporting, 64 

Pulley, winder, 47 to 54, 66 

Pulleys, wood, 56, 164 

, with double arms, 198 

, with steel arms, 185 

, very wide, 203 


Rawhide ropes, 25, 75 
Relative cost of ropes, 122 to 127 
Relative wear of ropes, 122 
Rim sections, 166, 187 
Ropes, cotton, 77, 81 

, cost of, 101, 122 

, early use of, 2 

, elasticity of, 162 

« I 

" , fibrous. 75 

" , flax, 77 









t ( 

, hemp, 2, 77, 90 
, horse-power of, 121 
, Lambeth, 80, 88 
, life of, 107 
, lubrication of, 79 
, manilla, 77 
, rawhide, 25, 75 
, round leather, 76 
, shrinkage of, 31 
, speed of, 103 
, splicing of, 18 

228 INDEX. 

Ropes, steel and leather, 76 

** , stevedore, 90 

** , strength of, 79, 91 

** , square leather, 76 

*• , wear of, 101, 104, 123 

" , weight of, 109 

** , wire, 75 
Rope wells, 10 
Rotation of ropes, 174 

Sag of ropes, 131, 170 
Sections of arms, 199 

" rim. 187 
Section of rope, 92 
Semicircular grooves, 164 
Sewall, M. W., built-up pulleys, 217 
Shafting, for long-distance transmissions, 68 

* * friction of, 145 . 

'* limit of length, 69 

'* loss due to friction, 68 

** power transmitted, 153 

Shafts, jack, use of, for dynamo-drives, 38, 42 
at an acute angle, 3^ 
at right angles, 36 
Shrinkage of ropes, 31 
Side lead of ropes, 35 
Size of pulley, 38 

" " ropes in use, 100, 107 
Slack-side tension, 111 
Small ropes, 24 
Speed of ropes, 103 
Split hubs, 191 
Splicing of ropes, 18 
Stevedore, transmission rope, 90 
Stress in rim-bolts, 196 

" ** ropes, 67 
Strength of ropes, 79, 91 
Supporting pulleys, 64 
Surface of groove, 183 

Taper of arms, 203 

INDEX. 229 


« ( 


Tallow on ropes, 89 
Tar, effect of, on ropes, 91 
Telodynamic transmissions, 63 
Temporary installations, 52 
Tension carriage, 27 

weight, 28, 32, 140 
in ropes, 111 to 118, 133 
Tests on ropes, 80 
Tightener, 29. 30 

" pulley, 48 

Transmissions at an angle, 34 

outdoor, 50, 66 

of power to a distance, 62 

telodynamic, 63 
Turbines, Victor, 56 
Twist in ropes, 86, 88 

Use of ropes with portable tools, 62 
Use of water-wheels, 56, 66, 71 

Van VJeck, built-up pulley, 214 
Vibration in ropes, 105 
Victor turbine. 56 

Walker, differential pulley, 173 
Walker Mfg. Co., built-up pulley, 210, 220 
Watertown tests on ropes, 80 
Water-wheels and rope-driving, 56, 66 
Wear of ropes, 83, 101, 104, 122 
Wear due to side lead, 35 
Weakening effect due to twisting, 86 
Webber, Samuel, early use of ropes, 2 

friction of shafting, 8 
Wedging action, 162, 169 
Weight for tension-carriage, 139 
Weight of ropes, 109 
Wells, rope, 10 

Western Electric Co. 'a plant, 25 
Wetherill & Co., built-up pulley, 221 
Wide pulleys, 203 
Willamette Mills, 66 
Winder-pulley, 47 to 54, 66 

5^30 INDEX. 

Wire ropes, 75 

Wood pulleys, 56, 164, 186 

Wood -filled pulley rim, 186 

WorkiDg strength of ropes, 95, 98 

Wound system of rope-driving, 25 

Wren's "Instrument for drawing up great weights," 45 


Bn>ln». EsUbllshHl . 45 BROADWAV, 

ID i87>. NEW VORK, 



This rope Is intended to be used 

»iL„ any otiier rope in tbe market, 
without any ejoeptlon whalever, 
and will e]aA\y refund the differeno- 

Yon cBDiiot liifle by making h trial 
of this rope, and are sure to reduce 
your expense account. Thia goar- 
antee goes witii every rope we 
make. We do not mannfactuce 
tranaroisaion poUeyB, K h tt f t i ng, or 
hangers, neither do we make ordiuary rope, 


zi}^ Inches gauge, is [ 
signed for use 

makes a "perfect permanent | ja™Si"iii ibb*™^ 


Por hnlHiirvB Tniii. firp. onrt stnne from veaselB, 
. and GafC hoistl 


, Hoii 

il Yards, on Whf. 

Ing Establishments. Coal Tubs. Steam Shovels, 
Mast and Gaff Fittings, Wheel Barrows, and 
Hoisting Blocks for both Manila and Wi "^ 





plant that will unload a 3,000 ton 
? Engines. Automatic and Cable 
itribnting Coal and Bulk Materiata 
dBtMannfactur- jb 

>. Steam Shovels, _^flM> 
el Barrows, and K r" t 
land Wire Rope, ^^^^^^fc 

ich carry the ^^^^^^V 
.age. or vfolencc ^^^^B^ 

1 Coaling StatiouB, ( 






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